Matrices

Q68 ……….matrix is both symmetric and skew-symmetric matrix. FillBlank Q69 Sum of two skew-symmetric matrices is always………matrix.
FillBlank Q70 The negative of a matrix is obtained by multiplying it by ………..
FillBlank Q71 The product of any matrix by the scalar ……….. is the null matrix.
FillBlank Q72 A matrix which is not a square matrix is called a ………….. matrix. FillBlank Q73 Matrix multiplication is ………… over addition.
FillBlank Q74 If A is a symmetric matrix, then ${A^3}$ is a ……….. matrix.
FillBlank Q75 If A is a skew-symmetric matrix, then ${A^2}$ is a …………..
FillBlank Q76 If A and B are square matrices of the same order, then
(i) ${(AB)^\prime } =$............
(ii) ${(kA)^\prime } =$......... (where, $k$ is any scalar)
(iii) ${[k(A - B)]^\prime } =$............ FillBlank Q77 If A is a skew-symmetric, then ${\rm{kA}}$ is a………… (where, $k$is any scalar).

FillBlank Q78 If A and B are symmetric matrices, then
(i) $AB - BA$ is a…………
(ii) $BA - 2AB$ is…………. FillBlank Q79 If A is symmetric matrix, then ${B^\prime }AB$ is…………
FillBlank Q80 If A and B are symmetric matrices of same order, then AB is symmetric if and only if..........
FillBlank Q81 In applying one or more row operations while finding ${A^{ - 1}}$ by elementary row operations, we obtain all zeroes in one or more, then ${A^{ - 1}}$..........
FillBlank Q49 If $AB = BA$ for any two square matrices, then prove by mathematical induction that ${(AB)^n} = {A^n}{B^n}$.
LA Q50 Find $x,y$ and $z$, if $A = \left[ {\begin{array}{cccccccccccccccccccc}0&{2y}&z\\x&y&{ - z}\\x&{ - y}&z\end{array}} \right]$ satisfies ${A^\prime } = {A^{ - 1}}$.

LA Q51 If possible, using elementary row transformations, find the inverse of the following matrices.

(i) $\left[ {\begin{array}{cccccccccccccccccccc}2&{ - 1}&3\\{ - 5}&3&1\\{ - 3}&2&3\end{array}} \right]$

(ii) $\left[ {\begin{array}{cccccccccccccccccccc}2&3&{ - 3}\\{ - 1}&{ - 2}&2\\1&1&{ - 1}\end{array}} \right]$

(iii) $\left[ {\begin{array}{cccccccccccccccccccc}2&0&{ - 1}\\5&1&0\\0&1&3\end{array}} \right]$
LA Q52 Express the matrix $\left[ {\begin{array}{cccccccccccccccccccc}2&3&1\\1&{ - 1}&2\\4&1&2\end{array}} \right]$ as the sum of a symmetric and a skew-symmetric matrix.

LA Q53 The matrix $P = \left[ {\begin{array}{llllllllllllllllllll}0&0&4\\0&4&0\\4&0&0\end{array}} \right]$ is a MCQ Q54 Total number of possible matrices of order $3 \times 3$ with each entry 2 or 0 is MCQ Q55 $\left[ {\begin{array}{cccccccccccccccccccc}{2x + y}&{4x}\\{5x - 7}&{4x}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}7&{7y - 13}\\y&{x + 6}\end{array}} \right]$, then the value of $x + y$ is MCQ Q56 If $A = \frac{1}{\pi }\left[ {\begin{array}{llllllllllllllllllll}{{{\sin }^{ - 1}}(x\pi )}&{{{\tan }^{ - 1}}\left( {\frac{x}{\pi }} \right)}\\{{{\sin }^{ - 1}}\left( {\frac{x}{\pi }} \right)}&{{{\cot }^{ - 1}}(\pi x)}\end{array}} \right]$ and $B = \frac{1}{\pi }\left[ {\begin{array}{llllllllllllllllllll}{ - {{\cos }^{ - 1}}(x\pi )}&{{{\tan }^{ - 1}}\left( {\frac{x}{\pi }} \right)}\\{{{\sin }^{ - 1}}\left( {\frac{x}{\pi }} \right)}&{ - {{\tan }^{ - 1}}(\pi x)}\end{array}} \right]$, then $A - B$ is equal to MCQ Q57 If A and B are two matrices of the order $3 \times {\rm{m}}$ and $3 \times n$, respectively and $m = n$, then order of matrix $(5A - 2B)$ is MCQ Q58 If $A = \left[ {\begin{array}{llllllllllllllllllll}0&1\\1&0\end{array}} \right]$, then ${A^2}$ is equal to MCQ Q59 If matrix $A = {\left[ {{a_{ij}}} \right]_{2 \times 2}}$, where ${a_{ij}} = 1$, if $i \ne j = 0$ and if $i = j$, then ${A^2}$ is equal to MCQ Q60 The matrix $\left[ {\begin{array}{llllllllllllllllllll}1&0&0\\0&2&0\\0&0&4\end{array}} \right]$ is a MCQ Q61 The matrix $\left[ {\begin{array}{cccccccccccccccccccc}0&{ - 5}&8\\5&0&{12}\\{ - 8}&{ - 12}&0\end{array}} \right]$ is a MCQ Q62 If A is matrix of order $m \times n$ and B is a matrix such that $A{B^\prime }$ and ${B^\prime }A$ are both defined, then order of matrix B is MCQ Q63 If A and B are matrices of same order, then $\left( {A{B^\prime } - B{A^\prime }} \right)$ is A MCQ Q64 If A is a square matrix such that ${A^2} = I$, then ${(A - I)^3} + {(A + I)^3} - 7A$ is equal to MCQ Q65 For any two matrices A and B, we have MCQ Q66 On using elementary column operations ${C_2} \to {C_2} - 2{C_1}$ in the following matrix equation $\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 3}\\2&4\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}\\0&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}3&1\\2&4\end{array}} \right]$, we have MCQ Q67 On using elementary row operation ${R_1} \to {R_1} - 3{R_2}$ in the following matrix equation $\left[ {\begin{array}{llllllllllllllllllll}4&2\\3&3\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}1&2\\0&3\end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll}2&0\\1&1\end{array}} \right]$, we have MCQ Q1 If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

SA Q2 In the matrix $A = \left[ {\begin{array}{cccccccccccccccccccc}a&1&x\\2&{\sqrt 3 }&{{x^2} - y}\\0&5&{\frac{{ - 2}}{5}}\end{array}} \right]$, write

(i) the order of the matrix A.

(ii) the number of elements.

(iii) elements ${a_{23}},{a_{31}}$ and ${a_{12}}$
SA Q3 Construct ${a_{2 \times 2}}$ matrix, where
(i) ${a_{ij}} = \frac{{{{(i - 2j)}^2}}}{2}$
(ii) ${a_{ij}} = | - 2i + 3j|$
SA Q4 Construct a $3 \times 2$ matrix whose elements are given by ${a_{ij}} = {e^{i \cdot x}} = \sin jx$.
SA Q5 Find the values of $a$ and $b$, if $A = B$, where
$A = \left[ {\begin{array}{cccccccccccccccccccc}{a + 4}&{3b}\\8&{ - 6}\end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc}{2a + 2}&{{b^2} + 2}\\8&{{b^2} - 5b}\end{array}} \right]$

SA Q6 If possible, find the sum of the matrices A and B, where $A = \left[ {\begin{array}{cccccccccccccccccccc}{\sqrt 3 }&1\\2&3\end{array}} \right]$ and $B = \left[ {\begin{array}{llllllllllllllllllll}x&y&z\\a&b&c\end{array}} \right]$.

SA Q7 If $X = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&1&{ - 1}\\5&{ - 2}&{ - 3}\end{array}} \right]$ and $Y = \left[ {\begin{array}{cccccccccccccccccccc}2&1&{ - 1}\\7&2&4\end{array}} \right]$, then find
(i) $X + Y$.
(ii) $2X - 3Y$.
(iii) a matrix $Z$ such that $X + Y + Z$ is a zero matrix.

SA Q8 Find non-zero values of $x$ satisfying the matrix equation $x\left[ {\begin{array}{cccccccccccccccccccc}{2x}&2\\3&x\end{array}} \right] + 2\left[ {\begin{array}{cccccccccccccccccccc}8&{5x}\\4&{4x}\end{array}} \right] = 2\left[ {\begin{array}{cccccccccccccccccccc}{\left( {{x^2} + 8} \right)}&{24}\\{(10)}&{6x}\end{array}} \right]$.

SA Q9 If $A = \left[ {\begin{array}{llllllllllllllllllll}0&1\\1&1\end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc}0&{ - 1}\\1&0\end{array}} \right]$, then show that
$(A + B)(A - B) \ne {A^2} - {B^2}$

SA Q10 Find the value of $x$, if $[1\,x\,1]\left[ {\begin{array}{cccccccccccccccccccc}1&3&2\\2&5&1\\{15}&3&2\end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll}1\\2\\x\end{array}} \right] = 0$.

SA Q11 Show that $A = \left[ {\begin{array}{cccccccccccccccccccc}5&3\\{ - 1}&{ - 2}\end{array}} \right]$

satisfies the equation ${A^2} - 3A - 7I = 0$ and hence find the value of ${A^{ - 1}}$.

SA Q12 Find the matrix A satisfying the matrix equation
$\left[ {\begin{array}{llllllllllllllllllll}2&1\\3&2\end{array}} \right]A\left[ {\begin{array}{cccccccccccccccccccc}{ - 3}&2\\5&{ - 3}\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}1&0\\0&1\end{array}} \right]$.
SA Q13 Find A, if $\left[ {\begin{array}{llllllllllllllllllll}4\\1\\3\end{array}} \right]A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 4}&8&4\\{ - 1}&2&1\\{ - 3}&6&3\end{array}} \right]$.
SA Q14 If $A\left[ {\begin{array}{cccccccccccccccccccc}3&{ - 4}\\1&1\\2&0\end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc}2&1&2\\1&2&4\end{array}} \right]$, then verify ${(BA)^2} \ne {B^2}{A^2}$.

SA Q15 If possible, find the value of BA and AB, where
$A = \left[ {\begin{array}{llllllllllllllllllll}2&1&2\\1&2&4\end{array}} \right]$ and $B = \left[ {\begin{array}{llllllllllllllllllll}4&1\\2&3\\1&2\end{array}} \right]$.

SA Q16 Show by an example that for $A \ne 0,B \ne 0$ and $AB = 0$.
SA Q17 Given, $A = \left[ {\begin{array}{llllllllllllllllllll}2&4&0\\3&9&6\end{array}} \right]$ and $B = \left[ {\begin{array}{llllllllllllllllllll}1&4\\2&8\\1&3\end{array}} \right]$. is ${(AB)^\prime } = {B^\prime }{A^\prime }$ ?
SA Q18 Solve for $x$ and $y$,$x\left[ {\begin{array}{llllllllllllllllllll}2\\1\end{array}} \right] + y\left[ {\begin{array}{llllllllllllllllllll}3\\5\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{ - 8}\\{ - 11}\end{array}} \right] = 0$.
SA Q19 If $X$ and $Y$ are $2 \times 2$ matrices, then solve the following matrix equations for $X$ and $Y$

$2X + 3Y = \left[ {\begin{array}{llllllllllllllllllll}2&3\\4&0\end{array}} \right]$, $3X + 2Y = \left[ {\begin{array}{cccccccccccccccccccc}{ - 2}&2\\1&{ - 5}\end{array}} \right]$. SA Q20 If $A = [3\,\,\,\,5]$ and $B = [7\,\,\,\,3]$, then find a non-zero matrix $C$ such that $AC = BC$.
SA Q21 Give an example of matrices A, B and C, such that $AB = AC,$where A is non-zero matrix but $B \ne C$.

SA Q22 If $A = \left[ {\begin{array}{cccccccccccccccccccc}1&2\\{ - 2}&1\end{array}} \right],B = \left[ {\begin{array}{cccccccccccccccccccc}2&3\\3&{ - 4}\end{array}} \right]$ and $C = \left[ {\begin{array}{cccccccccccccccccccc}1&0\\{ - 1}&0\end{array}} \right]$, verify

(i) $(AB)C = A(BC)$.

(ii) $A(B + C) = AB + AC$.
SA Q23 If $P = \left[ {\begin{array}{llllllllllllllllllll}x&0&0\\0&y&0\\0&0&z\end{array}} \right]$ and $Q = \left[ {\begin{array}{llllllllllllllllllll}a&0&0\\0&b&0\\0&0&c\end{array}} \right]$,

then prove that $PQ = \left[ {\begin{array}{cccccccccccccccccccc}{x{\rm{a}}}&0&0\\0&{{\rm{yb}}}&0\\0&0&{{\rm{zc}}}\end{array}} \right] = QP$
SA Q24 If $\left[ {\begin{array}{llllllllllllllllllll}2&1&3\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&0&{ - 1}\\{ - 1}&1&0\\0&1&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1\\0\\{ - 1}\end{array}} \right] = A$, then find the value of A.
SA Q25 If $A = [2\,\,\,\,\,\,\,1],$ $B = \left[ {\begin{array}{llllllllllllllllllll}5&3&4\\8&7&6\end{array}} \right]$ and $C = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&2&1\\1&0&2\end{array}} \right]$, then verify that
$A(B + C) = (AB + AC)$.

SA Q26 If $A = \left[ {\begin{array}{cccccccccccccccccccc}1&0&{ - 1}\\2&1&3\\0&1&1\end{array}} \right]$, then verify that ${A^2} + A = (A + I)$, where $I$ is $3 \times 3$ unit matrix.

SA Q27 If $A = \left[ {\begin{array}{cccccccccccccccccccc}0&{ - 1}&2\\4&3&{ - 4}\end{array}} \right]$ and $B = \left[ {\begin{array}{llllllllllllllllllll}4&0\\1&3\\2&6\end{array}} \right]$, then verify that
(i) ${\left( {{A^\prime }} \right)^\prime } = A$
(ii) ${(AB)^\prime } = {B^\prime }{A^\prime }$
(iii) ${(kA)^\prime } = \left( {k{A^\prime }} \right)$.

SA Q28 If $A = \left[ {\begin{array}{llllllllllllllllllll}1&2\\4&1\\5&6\end{array}} \right]$ and $B = \left[ {\begin{array}{llllllllllllllllllll}1&2\\6&4\\7&3\end{array}} \right]$, then verify that

(i) ${(2A + B)^\prime } = 2A\;{\rm{A}} + {B^\prime }$

(ii) ${(A - B)^\prime } = {A^\prime } - {B^\prime }$.

SA Q29 Show that ${A^\prime }A$ and $A{A^\prime }$ are both symmetric matrices for any matrix A.

SA Q30 Let A and B be square matrices of the order $3 \times 3$. Is ${(AB)^2} = {A^2}{B^2}?$ Give reasons.
SA Q31 Show that, if A and B are square matrices such that $AB = BA$, then ${(A + B)^2} = {A^2} + 2AB + {B^2}$.
SA Q32 If $A = \left[ {\begin{array}{cccccccccccccccccccc}1&2\\{ - 1}&3\end{array}} \right],B = \left[ {\begin{array}{llllllllllllllllllll}4&0\\1&5\end{array}} \right],C = \left[ {\begin{array}{cccccccccccccccccccc}2&0\\1&{ - 2}\end{array}} \right],a = 4$, and $b = - 2$, then show that
(i) $A + (B + C) = (A + B) + C$

(ii) $A(BC) = (AB)C$

(iii) $(a + b)B = aB + bB$

(iv) $a(C - A) = aC - aA$

(v) ${\left( {{A^T}} \right)^T} = A$

(vi) ${(bA)^T} = b{A^T}$

(vii) ${(AB)^T} = {B^T}{A^T}$

(viii) $(A - B)C = AC - BC$

(ix) ${(A - B)^T} = {A^T} - {B^T}$

SA Q33 If $A = \left[ {\begin{array}{cccccccccccccccccccc}{\cos q}&{\sin q}\\{ - \sin q}&{\cos q}\end{array}} \right]$, then show that ${A^2} = \left[ {\begin{array}{cccccccccccccccccccc}{\cos 2q}&{\sin 2q}\\{ - \sin 2q}&{\cos 2q}\end{array}} \right]$.
SA Q34 If $A = \left[ {\begin{array}{cccccccccccccccccccc}0&{ - x}\\x&0\end{array}} \right],B = \left[ {\begin{array}{cccccccccccccccccccc}0&1\\1&0\end{array}} \right]$ and ${x^2} = - 1$, then show that ${(A + B)^2} = {A^2} + {B^2}$.

SA Q35 Verify that ${A^2} = I$, when $A = \left[ {\begin{array}{cccccccccccccccccccc}0&1&{ - 1}\\4&{ - 3}&4\\3&{ - 3}&4\end{array}} \right]$.
SA Q36 Prove by mathematical induction that ${\left( {{A^\prime }} \right)^n} = {\left( {{A^n}} \right)^\prime }$ where $n \in N$ for any square matrix A.
SA Q37 Find inverse, by elementary row operations (if possible), of the following matrices.
(i) $\left[ {\begin{array}{cccccccccccccccccccc}1&3\\{ - 5}&7\end{array}} \right]$
(ii) $\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 3}\\{ - 2}&6\end{array}} \right]$

SA Q38 If $\left[ {\begin{array}{cccccccccccccccccccc}{xy}&4\\{z + 6}&{x + y}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}8&w\\0&6\end{array}} \right]$, then find the values of $x,y,z$ and $w$.

SA Q39 If $A = \left[ {\begin{array}{cccccccccccccccccccc}1&5\\7&{12}\end{array}} \right]$ and $B = \left[ {\begin{array}{llllllllllllllllllll}9&1\\7&8\end{array}} \right]$, then find a matrix C such that $3A + 5B + 2C$ is a null matrix.
SA Q40 If $A = \left[ {\begin{array}{cccccccccccccccccccc}3&{ - 5}\\{ - 4}&2\end{array}} \right]$, then find ${A^2} - 5A - 14I$. Hence, obtain ${A^3}$.

SA Q41 Find the values of $a,b,c$ and $d$, if
$3\left[ {\begin{array}{llllllllllllllllllll}a&b\\c&d\end{array}} \right] = \left[ \begin{array}{l}a\,\,\,\,\,\,\,\,\,\,6\\ - 1\,\,\,\,\,\,\,2d\end{array} \right] + \left[ {\begin{array}{cccccccccccccccccccc}4&{a + b}\\{c + d}&3\end{array}} \right]z$.
SA Q42 Find the matrix $A$ such that $\left[ {\begin{array}{cccccccccccccccccccc}2&{ - 1}\\1&0\\{ - 3}&4\end{array}} \right]A = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&{ - 8}&{ - 10}\\1&{ - 2}&{ - 5}\\9&{22}&{15}\end{array}} \right]$.
SA Q43 If $A = \left[ {\begin{array}{llllllllllllllllllll}1&2\\4&1\end{array}} \right]$, then find ${A^2} + 2A + 7I$.
SA Q44 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{\cos \alpha }&{\sin \alpha }\\{ - \sin \alpha }&{\cos \alpha }\end{array}} \right]$ and ${A^{ - 1}} = {A^\prime }$, then find the value of $\alpha$.

SA Q45 If matrix $\left[ {\begin{array}{cccccccccccccccccccc}0&a&3\\2&b&{ - 1}\\c&1&0\end{array}} \right]$ is a skew-symmetric matrix, then find the values of $a,b$ and $c$.

SA Q46 If $P(x) = \left[ {\begin{array}{llllllllllllllllllll}{\cos x}&{\sin x}\\{ - \sin x}&{\cos x}\end{array}} \right]$, then show that $P(x) \cdot P(y) = P(x + y)$ $= P(y) \cdot P(x)$.
SA Q47 If $A$ is square matrix such that ${A^2} = A$, then show that ${(I + A)^3} = 7A + I$.
SA Q48 If $A$, $B$ are square matrices of same order and $B$ is a skew-symmetric matrix, then show that ${A^\prime }BA$ is skew-symmetric.
SA Q83 Matrices of any order can be added.

Correct Answer False TrueFalse Q84 Two matrices are equal, if they have same number of rows and same number of columns.

Correct Answer False TrueFalse Q85 Matrices of different order cannot be subtracted.

Correct Answer True TrueFalse Q87 Matrix multiplication is commutative.

Correct Answer False TrueFalse Q88 A square matrix where every element is unity is called an identity matrix.

Correct Answer False TrueFalse Q89 If A and B are two square matrices of the same order, then A+B=B+A.

Correct Answer True TrueFalse Q90 If A and B are two matrices of the same order, then $A - B = B - A$.

Correct Answer False TrueFalse Q91 If matrix $AB = 0$, then $A = 0$ or $B = 0$ or both A and B are null matrices.

Correct Answer False TrueFalse Q92 Transpose of a column matrix is a column matrix.

Correct Answer False TrueFalse Q93 If A and B are two square matrices of the same order, then $AB = BA$.

Correct Answer False TrueFalse Q94 If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

Correct Answer True TrueFalse Q95 If A and B are any two matrices of the same order, then ${(AB)^\prime } = {A^\prime }{B^\prime }$.

Correct Answer False TrueFalse Q96 If ${(AB)^\prime } = {B^\prime }{A^\prime }$, where A and B are not square matrices,

then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.

Correct Answer True TrueFalse Q97 If A, B and C are square matrices of same order, then AB=AC always implies that B=C.

Correct Answer False TrueFalse Q98 $A{A^\prime }$ is always a symmetric matrix for any matrix A.

Correct Answer True TrueFalse Q99 If $A = \left| {\begin{array}{cccccccccccccccccccc}2&3&{ - 1}\\1&4&2\end{array}} \right|$ and $B = \left| {\begin{array}{cccccccccccccccccccc}2&3\\4&5\\2&1\end{array}} \right|$, then AB and BA are defined and equal.

Correct Answer False TrueFalse Q100 If A is skew-symmetric matrix, then ${A^2}$ is a symmetric matrix.

Correct Answer True TrueFalse Q101 ${(AB)^{ - 1}} = {A^{ - 1}} \cdot {B^{ - 1}}$, where A and B are invertible matrices satisfying commutative property with respect to multiplication.

Correct Answer True TrueFalse

Q1 In the matrix $A = \left[ {\begin{array}{cccccccccccccccccccc}2&5&{19}&{ - 7}\\{35}&{ - 2}&{5/2}&{12}\\{\sqrt 3 }&1&{ - 5}&{17}\end{array}} \right],$write :

• The order of the matrix,

• The number of elements,

• Write the elements ${a_{13}},{a_{21}},{a_{33}},{a_{24}},{a_{23}}.$

SA Q2 If a matrix has 24 elements, what are the possible orders it can have? What if, it has 13 elements?
SA Q3 If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
SA Q4 Construct a 2 × 2 matrix, $A = [{a_{ij}}]$, whose elements are given by :

• ${a_{ij}} = \cfrac{{{{(i + j)}^2}}}{2}$

• ${a_{ij}} = \cfrac{i}{j}$

• ${a_{ij}} = \cfrac{{{{(i + 2j)}^2}}}{2}$

SA Q5 Construct a 3 x 4 matrix, whose elements are given by :

• ${a_{ij}} = \cfrac{1}{2}| - 3i + j|$

• ${a_{ij}} = 2i - j$

SA Q6 Find the values of x, y and z from the following equations:

• $\left[ {\begin{array}{cccccccccccccccccccc}4&3\\x&5\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}y&z\\1&5\end{array}} \right]$

• $\left[ {\begin{array}{cccccccccccccccccccc}{x + y}&2\\{5 + z}&{xy}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}6&2\\5&8\end{array}} \right]$

• $\left[ {\begin{array}{cccccccccccccccccccc}{x + y + z}\\{x + z}\\{y + z}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}9\\5\\7\end{array}} \right]$

SA Q7 Find the values of a, b, c and d from the equation :

$\left[ {\begin{array}{cccccccccccccccccccc}{a - b}&{2a + c}\\{2a - b}&{3c + d}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&5\\0&{13}\end{array}} \right]$

SA Q8 $A = {[{a_{ij}}]_{m \times n}}$is a square matrix, if

• $m < n$

• $m> n$

• m = n

• None of these

SA Q9 Which of the given values of x and y make the following pair of matrices equal?

$\left[ {\begin{array}{cccccccccccccccccccc}{3x + 7}&5\\{y + 1}&{2 - 3x}\end{array}} \right],\left[ {\begin{array}{cccccccccccccccccccc}0&{y - 2}\\8&4\end{array}} \right]$

• $x = \cfrac{{ - 1}}{3},y = 7$

• Not possible to find

• $y = 7,x = \cfrac{{ - 2}}{3}$

• $x = \cfrac{{ - 1}}{3},y = \cfrac{{ - 2}}{3}$

SA Q10 The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is :

• 27

• 18

• 81

• 512

SA

Q1 Let $A = \left[ {\begin{array}{cccccccccccccccccccc}2&4\\3&2\end{array}} \right],B = \left[ {\begin{array}{cccccccccccccccccccc}1&3\\{ - 2}&5\end{array}} \right],C = \left[ {\begin{array}{cccccccccccccccccccc}{ - 2}&5\\3&4\end{array}} \right]$

Find each of the following :

(i) A+ B

(ii) A$-$B

(iii) 3A$-$ C

(iv) AB

(v) BA
SA Q2 Compute the following s

(i) $\left[ {\begin{array}{cccccccccccccccccccc}a&b\\{ - b}&a\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}a&b\\b&a\end{array}} \right]$

(ii) $\left[ {\begin{array}{cccccccccccccccccccc}{{a^2} + {b^2}}&{{b^2} + {c^2}}\\{{a^2} + {c^2}}&{{a^2} + {b^2}}\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{2ab}&{2bc}\\{ - 2ac}&{ - 2ab}\end{array}} \right]$

(iii) $\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&4&{ - 6}\\8&5&{16}\\2&8&5\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{12}&7&6\\8&0&5\\3&2&4\end{array}} \right]$

(iv) $\left[ {\begin{array}{cccccccccccccccccccc}{{{\cos }^2}x}&{{{\sin }^2}x}\\{{{\sin }^2}x}&{{{\cos }^2}x}\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{{{\sin }^2}x}&{{{\cos }^2}x}\\{{{\cos }^2}x}&{{{\sin }^2}x}\end{array}} \right]$
SA Q3 Compute the following products.

(i)

$\left[ {\begin{array}{cccccccccccccccccccc}a&b\\{ - b}&a\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}a&{ - b}\\b&a\end{array}} \right]$

(ii)

$\left[ {\begin{array}{cccccccccccccccccccc}1\\2\\3\end{array}} \right][\begin{array}{cccccccccccccccccccc}2&3&4\end{array}]$

(iii)

$\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 2}\\2&3\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1&2&3\\2&3&1\end{array}} \right]$

(iv) $\left[ {\begin{array}{cccccccccccccccccccc}2&3&4\\3&4&5\\4&5&6\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 3}&5\\0&2&4\\3&0&5\end{array}} \right]$

(v) $\left[ {\begin{array}{cccccccccccccccccccc}2&1\\3&2\\{ - 1}&1\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1&0&1\\{ - 1}&2&1\end{array}} \right]$

(vi) $\left[ {\begin{array}{cccccccccccccccccccc}3&{ - 1}&3\\{ - 1}&0&2\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}2&{ - 3}\\1&0\\3&1\end{array}} \right]$

SA Q4 If $A = \left[ {\begin{array}{cccccccccccccccccccc}1&2&{ - 3}\\5&0&2\\1&{ - 1}&1\end{array}} \right],B = \left[ {\begin{array}{cccccccccccccccccccc}3&{ - 1}&2\\4&2&5\\2&0&3\end{array}} \right]$ and $C = \left[ {\begin{array}{cccccccccccccccccccc}4&1&2\\0&3&2\\1&{ - 2}&3\end{array}} \right]$, then compute (A + B) and (B$-$C).
Also, verify that A + (B$-$C) = (A + B)$-$C.

SA Q5 If $A + \left[ {\begin{array}{cccccccccccccccccccc}{\cfrac{2}{3}}&1&{\cfrac{5}{3}}\\{\cfrac{1}{3}}&{\cfrac{2}{3}}&{\cfrac{4}{3}}\\{\cfrac{7}{3}}&2&{\cfrac{2}{3}}\end{array}} \right]$and $B = \left[ {\begin{array}{cccccccccccccccccccc}{\cfrac{2}{5}}&{\cfrac{3}{5}}&1\\{\cfrac{1}{5}}&{\cfrac{2}{5}}&{\cfrac{4}{5}}\\{\cfrac{7}{5}}&{\cfrac{6}{5}}&{\cfrac{2}{5}}\end{array}} \right],$ then compute $3A - 5B.$

SA Q6 Simplify,$\cos \theta \left[ {\begin{array}{cccccccccccccccccccc}{\cos \theta }&{\sin \theta }\\{ - \sin \theta }&{\cos \theta }\end{array}} \right] + \sin \theta \left[ {\begin{array}{cccccccccccccccccccc}{\sin \theta }&{ - \cos \theta }\\{\cos \theta }&{\sin \theta }\end{array}} \right]$

SA Q7 Find X and Y, if

(i) $X + Y = \left[ {\begin{array}{cccccccccccccccccccc}7&0\\2&5\end{array}} \right]$and $X - Y = \left[ {\begin{array}{cccccccccccccccccccc}3&0\\0&3\end{array}} \right]$

(ii) $2X + 3Y = \left[ {\begin{array}{cccccccccccccccccccc}2&3\\4&0\end{array}} \right]$ and $3X + 2Y = \left[ {\begin{array}{cccccccccccccccccccc}2&{ - 2}\\{ - 1}&5\end{array}} \right]$

SA Q8 Find X, if $Y = \left[ {\begin{array}{cccccccccccccccccccc}3&2\\1&4\end{array}} \right]$ and $2X + Y = \left[ {\begin{array}{cccccccccccccccccccc}1&0\\{ - 3}&2\end{array}} \right]$

SA Q9 Find x and y, if $2\left[ {\begin{array}{cccccccccccccccccccc}1&3\\0&x\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}y&0\\1&2\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}5&6\\1&8\end{array}} \right]$

SA Q10 Solve the equation for x, y, z and t,
if

$2\left[ {\begin{array}{cccccccccccccccccccc}x&z\\y&t\end{array}} \right] + 3\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}\\0&2\end{array}} \right] = 3\left[ {\begin{array}{cccccccccccccccccccc}3&5\\4&6\end{array}} \right]$
SA Q11 If $x\left[ {\begin{array}{cccccccccccccccccccc}2\\3\end{array}} \right] + y\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}\\1\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{10}\\5\end{array}} \right],$find the values of x and y.

SA Q12 Given $3\left[ {\begin{array}{cccccccccccccccccccc}x&y\\z&w\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}x&6\\{ - 1}&{2w}\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}4&{x + y}\\{z + w}&3\end{array}} \right]$,

find the values of x, y, z and w.

SA Q13 If $F(x) = \left[ {\begin{array}{cccccccccccccccccccc}{\cos x}&{ - \sin x}&0\\{\sin x}&{\cos x}&0\\0&0&1\end{array}} \right],$then show that F(x)$\cdot$F(y) = F(x + y).
SA Q14 Show that

(i)$\left[ {\begin{array}{cccccccccccccccccccc}5&{ - 1}\\6&7\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}2&1\\3&4\end{array}} \right] \ne \left[ {\begin{array}{cccccccccccccccccccc}2&1\\3&4\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}5&{ - 1}\\6&7\end{array}} \right]$

(ii) $\left[ {\begin{array}{cccccccccccccccccccc}1&2&3\\0&1&0\\1&1&0\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&1&0\\0&{ - 1}&1\\2&3&4\end{array}} \right] \ne \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&1&0\\0&{ - 1}&1\\2&3&4\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1&2&3\\0&1&0\\1&1&0\end{array}} \right]$
SA Q15 Find ${A^2} - 5A + 6I,$If $A = \left[ {\begin{array}{cccccccccccccccccccc}2&0&1\\2&1&3\\1&{ - 1}&0\end{array}} \right]$
SA Q16 If $A = \left[ {\begin{array}{cccccccccccccccccccc}1&0&2\\0&2&1\\2&0&3\end{array}} \right]$, prove that ${A^3} - 6{A^2} + 7A + 2I = O$.
SA Q17 If $A = \left[ {\begin{array}{cccccccccccccccccccc}3&{ - 2}\\4&{ - 2}\end{array}} \right]$ and $I = \left[ {\begin{array}{cccccccccccccccccccc}1&0\\0&1\end{array}} \right]$, find k so that ${A^2} = kA = 2I.$

SA Q18 If $A = \left[ {\begin{array}{cccccccccccccccccccc}0&{ - \tan \cfrac{\alpha }{2}}\\{\tan \cfrac{\alpha }{2}}&0\end{array}} \right]$ and I is the identity matrix of order 2,

then show that I + A = (I$-$ A)$\left[ {\begin{array}{cccccccccccccccccccc}{\cos \alpha }&{ - \sin \alpha }\\{cin\alpha }&{\cos \alpha }\end{array}} \right]$.

SA Q19 A trust fund has Rs. 30, 000 that must be invested in two different types of bonds. The first bond pays 5\% interest per year, and the second bond pays 7\% interest per year. Using matrix multiplication, determine how to divide Rs. 30, 000 among the two types of bonds, if the trust fund obtains an annual total interest of :

(A) Rs. 1800

(B) Rs. 2000
SA Q20 The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

SA Q21 The restriction on n, k and p so that PY + WY will be defined are $\cdots$

SA Q22 If n= p, then the order of the matrix 7X$-$5Z is

SA

Q1 Find the transpose of each of the following matrices :

(i) $\left[ {\begin{array}{cccccccccccccccccccc}5\\{\cfrac{1}{2}}\\{ - 1}\end{array}} \right]$

(ii) $\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}\\2&3\end{array}} \right]$

(iii) $\left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&5&6\\{\sqrt 3 }&5&6\\2&3&{ - 1}\end{array}} \right]$

SA Q2 If $A = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&2&3\\5&7&9\\{ - 2}&1&1\end{array}} \right]$and $B = \left[ {\begin{array}{cccccccccccccccccccc}{ - 4}&1&{ - 5}\\1&2&0\\1&3&1\end{array}} \right]$, then verify that

(i) $(A + B)' = A' + B',$

(ii) $(A - B)' = A' - B'$
SA Q3 If $A' = \left[ {\begin{array}{cccccccccccccccccccc}3&4\\{ - 1}&2\\0&1\end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&2&1\\1&2&3\end{array}} \right]$, then verify that

(i) $(A + B)' = A' + B'$

(ii) $(A - B)' = A' - B'$

SA Q4 If $A' = \left[ {\begin{array}{cccccccccccccccccccc}{ - 2}&3\\1&2\end{array}} \right]$and $B = \left[ {\begin{array}{cccccccccccccccccccc}{ - 1}&0\\1&2\end{array}} \right]$, then find $(A + 2B)'$.
SA Q5 For the matrices A and B, verify that (AB)’ =B’A’, where

(i) $A = \left[ {\begin{array}{cccccccccccccccccccc}1\\{ - 4}\\3\end{array}} \right],B = [\begin{array}{cccccccccccccccccccc}{ - 1}&2&{1]}\end{array}$

(ii) $A = \left[ {\begin{array}{cccccccccccccccccccc}0\\1\\2\end{array}} \right],B = [\begin{array}{cccccccccccccccccccc}1&5&{7]}\end{array}$
SA Q6 If

$(i)A = \left[ {\begin{array}{cccccccccccccccccccc}{\cos \alpha }&{\sin \alpha }\\{ - \sin \alpha }&{\cos \alpha }\end{array}} \right],$then verify that $A'A = I$

(ii) $A = \left[ {\begin{array}{cccccccccccccccccccc}{\sin \alpha }&{\cos \alpha }\\{ - \cos \alpha }&{\sin \alpha }\end{array}} \right],$ then verify that $A'A = I$

SA Q7 (i) Show that the matrix $A = \left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}&5\\{ - 1}&2&1\\5&1&3\end{array}} \right]$is a symmetric matrix.

(ii) Show that the matrix $A = \left[ {\begin{array}{cccccccccccccccccccc}0&1&{ - 1}\\{ - 1}&0&1\\1&{ - 1}&0\end{array}} \right]$is a skew symmetric matrix.

SA Q8 For the matrix $A = \left[ {\begin{array}{cccccccccccccccccccc}1&5\\6&7\end{array}} \right]$, verify that

(i) (A + A’) is a symmetric matrix.

(ii) (A$-$A’) is a skew-symmetric matrix.

SA Q9 Find (A+ A’) and $\cfrac{1}{2}(A - A')$, when , $A = \left[ {\begin{array}{cccccccccccccccccccc}0&a&b\\{ - a}&0&c\\{ - b}&{ - c}&0\end{array}} \right]$
SA Q10 Express the following matrices as the sum of a symmetric and a skew symmetric matrix :

(i) $\left[ {\begin{array}{cccccccccccccccccccc}3&5\\1&{ - 1}\end{array}} \right]$

(ii) $\left[ {\begin{array}{cccccccccccccccccccc}6&{ - 2}&2\\{ - 2}&3&{ - 1}\\2&{ - 1}&3\end{array}} \right]$

(iii)$\left[ {\begin{array}{cccccccccccccccccccc}3&3&{ - 1}\\{ - 2}&{ - 2}&1\\{ - 4}&{ - 5}&2\end{array}} \right]$

(iv)$\left[ {\begin{array}{cccccccccccccccccccc}1&5\\{ - 1}&2\end{array}} \right]$

SA Q11 If A, B are symmetric matrices of same order, then AB$-$BA is a

SA Q12 If $A = \left[ {\begin{array}{cccccccccccccccccccc}{\cos \alpha }&{ - \sin \alpha }\\{\sin \alpha }&{\cos \alpha }\end{array}} \right],$then $A + A' = I,$If the value of $\alpha$ is

SA

Q1 $\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}\\2&3\end{array}} \right]$

SA Q2 $\left[ {\begin{array}{cccccccccccccccccccc}2&1\\1&1\end{array}} \right]$
SA Q3 $\left[ {\begin{array}{cccccccccccccccccccc}1&3\\2&7\end{array}} \right]$

SA Q4 $\left[ {\begin{array}{cccccccccccccccccccc}2&3\\5&7\end{array}} \right]$

SA Q5 $\left[ {\begin{array}{cccccccccccccccccccc}2&1\\7&4\end{array}} \right]$
SA Q6 $\left[ {\begin{array}{cccccccccccccccccccc}2&5\\1&3\end{array}} \right]$

SA Q7 $\left[ {\begin{array}{cccccccccccccccccccc}3&1\\5&2\end{array}} \right]$
SA Q8 $\left[ {\begin{array}{cccccccccccccccccccc}4&5\\3&4\end{array}} \right]$
SA Q9 $\left[ {\begin{array}{cccccccccccccccccccc}3&{10}\\2&7\end{array}} \right]$
SA Q10 $\left[ {\begin{array}{cccccccccccccccccccc}3&{ - 1}\\{ - 4}&2\end{array}} \right]$

SA Q11 $\left[ {\begin{array}{cccccccccccccccccccc}2&{ - 6}\\1&{ - 2}\end{array}} \right]$
SA Q12 $\left[ {\begin{array}{cccccccccccccccccccc}6&{ - 3}\\{ - 2}&1\end{array}} \right]$

SA Q13 $\left[ {\begin{array}{cccccccccccccccccccc}2&{ - 3}\\{ - 1}&2\end{array}} \right]$

SA Q14 $\left[ {\begin{array}{cccccccccccccccccccc}2&1\\4&2\end{array}} \right]$

SA Q15 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 3}&3\\2&2&3\\3&{ - 2}&2\end{array}} \right]$

SA Q16 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&3&{ - 2}\\{ - 3}&0&{ - 5}\\2&5&0\end{array}} \right]$

SA Q17 $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&0&{ - 1}\\5&1&0\\0&1&3\end{array}} \right]$

SA Q18 Matrices A and B will be inverse of each other only if

SA

Q1 Let A = $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&1\\0&0\end{array}} \right]$, show that ${(aI + bA)^n} + {a^n}I + n{a^{n - 1}}bA,$where I is the identity matrix of order 2 and $n \in N.$

SA Q2 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&1\\1&1&1\\1&1&1\end{array}} \right]$, prove that ${A^n} = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{{3^{n - 1}}}&{{3^{n - 1}}}&{{3^{n - 1}}}\\{{3^{n - 1}}}&{{3^{n - 1}}}&{{3^{n - 1}}}\\{{3^{n - 1}}}&{{3^{n - 1}}}&{{3^{n - 1}}}\end{array}} \right]$, $n \in N.$

SA Q3 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&{ - 4}\\1&{ - 1}\end{array}} \right]$, then prove that ${A^n} = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{1 + 2n}&{ - 4n}\\n&{1 - 2n}\end{array}} \right],$where n is any positive integer.

SA Q4 If A and B are symmetric matrices, prove that AB$-$BA is a skew symmetric matrix.

SA Q5 . Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

SA Q6 Find the values of x, y, z if the matrix $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&{2y}&z\\x&y&{ - z}\\x&{ - y}&z\end{array}} \right]$ satisfy the equation $A'A = I.$

SA Q7 For what values of $x:\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2&1\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2&0\\2&0&1\\1&0&2\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0\\2\\x\end{array}} \right] = O?$

SA Q8 If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}3&1\\{ - 1}&2\end{array}} \right]$, then show that ${A^2} - 5A + 7I = O$.

SA Q9 Find x, if $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&{ - 5}&{ - 1}\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&0&2\\0&2&1\\2&0&3\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x\\4\\1\end{array}} \right] = O$.

SA Q10 A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated as :

(a) If unit sale prices of x,y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit

. SA Q11 Find the matrix X so that $X\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&2&3\\4&5&6\end{array}} \right] = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{ - 7}&{ - 8}&{ - 9}\\2&4&6\end{array}} \right]$

SA Q12 If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = BnA. Further, prove that ${(AB)^n} = {A^n}{B^n}$for all $n \in N$.

SA Q13 If$A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}\alpha &\beta \\\gamma &{ - \alpha }\end{array}} \right]$is such that $A^2 = I$, then

SA Q14 If the matrix A is both symmetric and skew symmetric, then

SA Q15 If A is a square matrix such that $A^2 = A$, then ${(I + A)^3} - 7A$is equal to

SA