Matrices — Honors Math practice questions

Question 1

The identity matrix $I_3$ is:

Accepted Answer

$I_3$ has $1$s on the main diagonal and $0$s elsewhere.

Answer

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

Answer

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

Answer

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

Answer

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

Question 2

The trace of $A = \begin{pmatrix} 2 & 7 & 1 \ 0 & 3 & 4 \ 1 & 0 & 5 \end{pmatrix}$ is:

Accepted Answer

$\operatorname{tr}(A) = 2 + 3 + 5 = 10$.

Answer

$5$

Answer

$9$

Answer

$10$

Answer

$23$

Question 3

For $A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$, the matrix $A^2$ equals:

Accepted Answer

$A^2 = \begin{pmatrix} 1 + 0 & 1 + 1 \ 0 + 0 & 0 + 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$.

Answer

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

Answer

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

Answer

$\begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$

Answer

$\begin{pmatrix} 2 & 2 \ 0 & 2 \end{pmatrix}$

Question 4

For $A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix}$, the transpose $A^T$ has dimensions:

Accepted Answer

$A$ is $2 	imes 3$, so $A^T$ is $3 	imes 2$.

Answer

$2 	imes 2$

Answer

$2 	imes 3$

Answer

$3 	imes 2$

Answer

$3 	imes 3$

Question 5

For general matrices $A$ and $B$ in $M_n(\mathbb{R})$ (with $n \ge 2$), which is true?

Accepted Answer

Matrix multiplication is **not** commutative in general. (Concrete example: $A = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$, $B = \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}$ — check $AB 
e BA$.)

Answer

$AB = BA$ always

Answer

$AB 
e BA$ in general

Answer

$AB$ does not exist

Answer

$A + B 
e B + A$

Question 6

For $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \ 1 & 4 \end{pmatrix}$, the product $AB$ equals:

Accepted Answer

$(AB)_{11} = 1 \cdot 3 + 2 \cdot 1 = 5$; $(AB)_{12} = 1 \cdot 0 + 2 \cdot 4 = 8$; $(AB)_{21} = 0 \cdot 3 + 1 \cdot 1 = 1$; $(AB)_{22} = 0 \cdot 0 + 1 \cdot 4 = 4$. So $AB = \begin{pmatrix} 5 & 8 \ 1 & 4 \end{pmatrix}$.

Answer

$\begin{pmatrix} 3 & 0 \ 0 & 4 \end{pmatrix}$

Answer

$\begin{pmatrix} 5 & 8 \ 1 & 4 \end{pmatrix}$

Answer

$\begin{pmatrix} 3 & 8 \ 1 & 4 \end{pmatrix}$

Answer

$\begin{pmatrix} 5 & 4 \ 3 & 4 \end{pmatrix}$

Question 7

For $A = \begin{pmatrix} 2 & -1 \ 0 & 3 \end{pmatrix}$, the matrix $3A$ equals:

Accepted Answer

Multiply each entry by $3$: $\begin{pmatrix} 6 & -3 \ 0 & 9 \end{pmatrix}$.

Answer

$\begin{pmatrix} 5 & 2 \ 3 & 6 \end{pmatrix}$

Answer

$\begin{pmatrix} 2 & -1 \ 0 & 9 \end{pmatrix}$

Answer

$\begin{pmatrix} 6 & -3 \ 0 & 9 \end{pmatrix}$

Answer

$\begin{pmatrix} 6 & -1 \ 0 & 9 \end{pmatrix}$

Question 8

For $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & -1 \ 5 & 2 \end{pmatrix}$, $A + B$ equals:

Accepted Answer

Add componentwise: $\begin{pmatrix} 1 & 1 \ 8 & 6 \end{pmatrix}$.

Answer

$\begin{pmatrix} 1 & 1 \ 8 & 6 \end{pmatrix}$

Answer

$\begin{pmatrix} 1 & 3 \ 8 & 2 \end{pmatrix}$

Answer

$\begin{pmatrix} 1 & -1 \ -2 & 2 \end{pmatrix}$

Answer

$\begin{pmatrix} 0 & -2 \ 15 & 8 \end{pmatrix}$

Question 9

For $A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$, the entry $(A^n)_{12}$ equals (for $n \ge 1$):

Accepted Answer

By induction, $A^n = \begin{pmatrix} 1 & n \ 0 & 1 \end{pmatrix}$, so $(A^n)_{12} = n$.

Answer

$1$

Answer

$n - 1$

Answer

$n$

Answer

$n^2$

Question 10

For which value of $a \in \mathbb{R}$ is $A = \begin{pmatrix} 1 & a \ 3 & 2 \end{pmatrix}$ symmetric?

Accepted Answer

We need $a_{12} = a_{21}$, i.e. $a = 3$.

Answer

$-3$

Answer

$0$

Answer

$3$

Answer

$1$

Matrices practice questions — Honors Math

Problems & worked solutions

Problem 1 — Matrices

Problem 2 — Matrices

Problem 3 — Matrices

Problem 4 — Matrices

Problem 5 — Matrices

Problem 6 — Matrices

Problem 7 — Matrices

Problem 8 — Matrices

Problem 9 — Matrices

Problem 10 — Matrices