Online Test — Matrices
10 Questions • 20 min • Chapter MCQ
20:00
Question 1 of 10
easy
If $A$ is a $3\times4$ matrix and $B$ is a $4\times2$ matrix, the order of $AB$ is:
$4\times4$
$3\times2$
$3\times4$
$4\times2$
Explanation: The result of multiplying an $m\times n$ matrix by an $n\times p$ matrix is $m\times p$. Here $3\times4$ times $4\times2$ gives $3\times2$.
Question 2 of 10
easy
A square matrix $A$ is skew-symmetric if:
$A=A'$
$A=-A'$
$A^2=A$
$A^{-1}=A'$
Explanation: A skew-symmetric matrix satisfies $A'=-A$ (equivalently $A=-A'$). Diagonal entries must all be 0.
Question 3 of 10
easy
If $A=\begin{pmatrix}2 & 1 \\ 3 & 2\end{pmatrix}$, then $A^{-1}$ is:
$\begin{pmatrix}2 & -1 \\ -3 & 2\end{pmatrix}$
$\begin{pmatrix}-2 & 1 \\ 3 & -2\end{pmatrix}$
$\begin{pmatrix}2 & 1 \\ 3 & 2\end{pmatrix}$
$\frac{1}{7}\begin{pmatrix}2 & -1 \\ -3 & 2\end{pmatrix}$
Explanation: $\det A=4-3=1$. $A^{-1}=\frac{1}{1}\begin{pmatrix}2&-1\\-3&2\end{pmatrix}$.
Question 4 of 10
medium
If $A$ is a symmetric matrix, then $A^3$ is:
Skew-symmetric
Symmetric
Zero matrix
Identity
Explanation: If $A'=A$, then $(A^3)' = (A')^3 = A^3$. So $A^3$ is also symmetric.
Question 5 of 10
medium
If $A = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$, then $A^{2026}$ is:
$A$
$I$
$-I$
$-A$
Explanation: $A^2 = \begin{pmatrix}1&0\\0&1\end{pmatrix} = I$. So $A^{2026} = (A^2)^{1013} = I^{1013} = I$.
Question 6 of 10
easy
The number of possible orders of a matrix having 8 elements is:
2
4
3
6
Explanation: Factors of 8: $1\times8,2\times4,4\times2,8\times1$ — 4 possible orders.
Question 7 of 10
hard
If $A$ and $B$ are square matrices of the same order and $AB = 0$, then:
$A=0$ or $B=0$
$A=0$ and $B=0$
It's possible for both $A,B$ to be non-zero
$A$ and $B$ commute
Explanation: Matrix multiplication is not like real numbers — $AB=0$ does not imply $A=0$ or $B=0$. Example: $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$, $B=\begin{pmatrix}0&0\\0&1\end{pmatrix}$, $AB=0$.
Question 8 of 10
hard
If $A$ is a $2\times2$ matrix with $A^2 = A$, then $(I+A)^3 = I + kA$ for $k = $:
7
3
1
8
Explanation: $(I+A)^3=I+3A+3A^2+A^3=I+3A+3A+A=I+7A$ (using $A^2=A$, $A^3=A$). So $k=7$.
Question 9 of 10
easy
$(AB)' =$ ?
$A'B'$
$B'A'$
$BA$
$A'B$
Explanation: By the transpose property of a product: $(AB)' = B'A'$. The order reverses.
Question 10 of 10
medium
A matrix is both symmetric and skew-symmetric if and only if it is:
Identity matrix
Diagonal matrix
Zero matrix
Scalar matrix
Explanation: If $A=A'$ (symmetric) and $A=-A'$ (skew-symmetric), then $A'=A=-A'$, so $2A'=0$, giving $A=0$.