IMO Practice Test: Matrices | Class 12 Mathematics

Question 1 of 6 hard

If $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, then $A^n$ equals:

$\begin{pmatrix}1&n\\0&1\end{pmatrix}$

$\begin{pmatrix}n&1\\0&n\end{pmatrix}$

$nA$

$\begin{pmatrix}1&0\\0&1\end{pmatrix}$

Explanation: By induction: $A^n=\begin{pmatrix}1&n\\0&1\end{pmatrix}$ (upper-triangular, diagonal 1s, upper entry $n$).

Question 2 of 6 hard

If $A$ is a $3\times3$ matrix with $A^3=0$, which statement is necessarily true?

$A=0$

$A^2=0$

$(I+A)$ is invertible

$(I-A)^{-1}=I$

Explanation: $(I+A)(I-A+A^2)=I-A^3=I-0=I$. So $(I+A)$ is invertible.

Question 3 of 6 hard

For a $2\times2$ matrix, if $A^2=I$ and $A\ne\pm I$, how many such matrices with integer entries in $\{0,1,-1\}$ exist?

4

6

8

12

Explanation: $A^2=I$ means $A=A^{-1}$. For $2\times2$ real matrices, solutions include $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$, $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$, $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ ... counting carefully in $\{0,1,-1\}^{2\times2}$: 6.

Question 4 of 6 hard

If $AB = A$ and $BA = B$, then $B^2 = $?

$B$

$A$

$I$

$0$

Explanation: $B^2 = B(BA) = (BB)A$... better: $B^2 = B\cdot B = (BA)B = B(AB) = BA = B$. So $B^2=B$ (B is idempotent).

Question 5 of 6 hard

The trace (sum of diagonal) of $AB - BA$ for any two $2\times2$ matrices $A,B$ is:

Always positive

Always negative

Always zero

Depends on A and B

Explanation: $\text{tr}(AB)=\text{tr}(BA)$ always. So $\text{tr}(AB-BA)=\text{tr}(AB)-\text{tr}(BA)=0$.

Question 6 of 6 medium

If $A$ is a $n\times n$ invertible matrix and $B$ is a $n\times m$ matrix, then the system $AX=B$:

Has no solution

Has a unique solution $X=A^{-1}B$

Has infinitely many solutions

May or may not have a solution

Explanation: Since $A$ is invertible, multiply both sides on the left by $A^{-1}$: $X=A^{-1}B$. Unique solution.

IMO Practice Test — Matrices