Q: Which of the following matrices is both diagonal and an identity matrix?

$I_3 = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

Q: Let $A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]$ and $B = \left[ \begin{array}{cc} 5 & 6 \\ 7 & 8 \end{array} \right]$. What is $A + B$?

$\left[ \begin{array}{cc} 6 & 8 \\ 10 & 12 \end{array} \right]$

Q: Given $A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]$, calculate $3A$.

$\left[ \begin{array}{cc} 3 & 6 \\ 9 & 12 \end{array} \right]$

Q: Consider the system of equations: $4x + 2y = 4$, $5x + y = 8$, $6x + 3y = 6$. Which matrix equation represents this system?

$\left[ \begin{array}{cc} 4 & 2 \\ 5 & 1 \\ 6 & 3 \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} 4 \\ 8 \\ 6 \end{array} \right]$

Question 1

Which of the following statements correctly defines a matrix?

Accepted Answer

A matrix is a rectangular array of numbers, usually real or complex, arranged in rows and columns.

Question 2

Given the matrix $A = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} ight]$, which of the following is the diagonal of $A$?

Accepted Answer

$[1, 4]$

Question 3

Which of the following matrices is both diagonal and an identity matrix?

Accepted Answer

$I_3 = \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} ight]$

Question 4

If $A$ is a $2 	imes 3$ matrix and $B$ is a $3 	imes 4$ matrix, what is the size of the product $AB$?

Accepted Answer

$2 	imes 4$

Question 5

Which of the following is NOT a property of matrix addition?

Accepted Answer

Matrix addition is defined for any two matrices, regardless of their sizes.

Question 6

Let $A = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} ight]$ and $B = \left[ \begin{array}{cc} 5 & 6 \ 7 & 8 \end{array} ight]$. What is $A + B$?

Accepted Answer

$\left[ \begin{array}{cc} 6 & 8 \ 10 & 12 \end{array} ight]$

Question 7

If $A$ is a $3 	imes 3$ diagonal matrix with all diagonal entries equal to $2$, what type of matrix is $A$?

Accepted Answer

A scalar matrix

Question 8

Suppose $A$ is a $2 	imes 2$ matrix and $\vec{v}$ is a column vector in $\mathbb{R}^2$. What is the result of $A\vec{v}$?

Accepted Answer

A column vector in $\mathbb{R}^2$

Question 9

Which of the following statements about matrix multiplication is TRUE?

Accepted Answer

The number of columns of the first matrix must equal the number of rows of the second matrix.

Question 10

Given $A = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} ight]$, calculate $3A$.

Accepted Answer

$\left[ \begin{array}{cc} 3 & 6 \ 9 & 12 \end{array} ight]$

Question 11

If $A$ is a $3 	imes 3$ matrix and $I_3$ is the $3 	imes 3$ identity matrix, what is $AI_3$?

Accepted Answer

$A$

Question 12

Which of the following is the definition of a zero matrix?

Accepted Answer

A matrix with all entries equal to zero.

Question 13

Consider the system of equations: $4x + 2y = 4$, $5x + y = 8$, $6x + 3y = 6$. Which matrix equation represents this system?

Accepted Answer

$\left[ \begin{array}{cc} 4 & 2 \ 5 & 1 \ 6 & 3 \end{array} ight] \left[ \begin{array}{c} x \ y \end{array} ight] = \left[ \begin{array}{c} 4 \ 8 \ 6 \end{array} ight]$

Question 14

If $A$ is a $n 	imes n$ matrix, which of the following expressions is always defined?

Accepted Answer

$A^2$

Question 15

Which of the following is TRUE about the powers of a square matrix $A$?

Accepted Answer

$A^k$ is defined as $A$ multiplied by itself $k$ times.

Question 16

Suppose $A$ is a $3 	imes 3$ matrix and $\vec{e}_1$ is the first standard unit vector in $\mathbb{R}^3$. What is $A\vec{e}_1$?

Accepted Answer

The first column of $A$

Question 17

True or False: Every diagonal matrix is a scalar matrix.

Accepted Answer

False

Question 18

True or False: If $A$ and $B$ are both square matrices, then $AB$ is square.

Accepted Answer

True

Question 19

Is it possible for a nonzero matrix $A$ to satisfy $A^2 = O$ (the zero matrix)?

Accepted Answer

Yes, for example $A = \left[ \begin{array}{cc} 0 & 1 \ 0 & 0 \end{array} ight]$.

Matrix Operations: Definitions, Properties, and Applications

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