Q: Let $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}. $$Find $$A + B.$$

$$\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$$

Q: If $$A = \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}$$, what is $$3A$$?

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

Q: Given $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$, find $$AB.$$

$$\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}$$

Q: Which of the following is NOT a valid operation for matrices $$A$$ and $$B$$?

Adding $$A$$ and $$B$$ when $$A is$$ $$2 \times 3$$ and $$B is$$ $$3 \times 2$$

Q: Let $$A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 4 \\ 2 & 5 \end{bmatrix}. $$Find $$A - B.$$

$$\begin{bmatrix} 1 & -4 \\ -1 & -2 \end{bmatrix}$$

Q: Suppose $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix}. $$Compute $$AB.$$

$$\begin{bmatrix} 4 & 4 \\ 10 & 8 \end{bmatrix}$$

Q: Let $$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$ and $$B = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}. $$Find the entry in the first row, second column of $$AB.$$

$$1 \times 8 + 2 \times 10 + 3 \times 12 = 8 + 20 + 36 = 64$$

Question 1

Given matrix $$A$$ has dimensions $$2 	imes 3$$ and matrix $$B$$ has dimensions $$3 	imes 4. $$What are the dimensions of the product $$AB$$?

Accepted Answer

$$2 	imes 4$$

Question 2

Which of the following statements is TRUE about matrix multiplication?

Accepted Answer

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

Question 3

Let $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}. $$Find $$A + B.$$

Accepted Answer

$$\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$$

Question 4

If $$A = \begin{bmatrix} 2 & -1 \ 0 & 3 \end{bmatrix}$$, what is $$3A$$?

Accepted Answer

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

Question 5

Given $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$$, find $$AB.$$

Accepted Answer

$$\begin{bmatrix} 2 & 1 \ 4 & 3 \end{bmatrix}$$

Question 6

Which of the following is NOT a valid operation for matrices $$A$$ and $$B$$?

Accepted Answer

Adding $$A$$ and $$B$$ when $$A is$$ $$2 	imes 3$$ and $$B is$$ $$3 	imes 2$$

Question 7

A company tracks sales in three regions over two months using the matrix $$S = \begin{bmatrix} 100 & 150 & 200 \ 120 & 130 & 140 \end{bmatrix}. $$What are the dimensions of $$S$$?

Accepted Answer

$$2 	imes 3$$

Question 8

Let $$A = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 4 \ 2 & 5 \end{bmatrix}. $$Find $$A - B.$$

Accepted Answer

$$\begin{bmatrix} 1 & -4 \ -1 & -2 \end{bmatrix}$$

Question 9

If $$A is a$$ $$3 	imes 2$$ matrix and $$B is a$$ $$2 	imes 5$$ matrix, what are the dimensions of $$AB$$?

Accepted Answer

$$3 	imes 5$$

Question 10

Suppose $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 0 \ 1 & 2 \end{bmatrix}. $$Compute $$AB.$$

Accepted Answer

$$\begin{bmatrix} 4 & 4 \ 10 & 8 \end{bmatrix}$$

Question 11

A hospital tracks the number of patients in three departments over four days using a matrix $$P. If$$ $$P$$ has 3 rows and 4 columns, what does each entry $$p_{ij}$$ represent?

Accepted Answer

The number of patients in department $$i on $$day $$j$$

Question 12

Let $$A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$$ and $$B = \begin{bmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{bmatrix}. $$Find the entry in the first row, second column of $$AB.$$

Accepted Answer

$$1 	imes 8 + 2 	imes 10 + 3 	imes 12 = 8 + 20 + 36 = 64$$

Question 13

If $$A is a$$ $$2 	imes 3$$ matrix and $$B is a$$ $$3 	imes 2$$ matrix, which of the following is true?

Accepted Answer

Both $$AB$$ and $$BA$$ are defined, but have different dimensions.

Question 14

A scientist records the temperature in four cities over three days in a matrix $$T$$ with dimensions $$4 	imes 3. $$What does the entry $$t_{23}$$ represent?

Accepted Answer

The temperature in city 2 on day 3

Question 15

Let $$A = \begin{bmatrix} 2 & 1 \ 0 & 3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 2 \ 4 & 0 \end{bmatrix}. $$Find $$AB.$$

Accepted Answer

$$\begin{bmatrix} 6 & 4 \ 12 & 0 \end{bmatrix}$$

Question 16

Which of the following best describes the process of multiplying two matrices $$A$$ and $$B$$?

Accepted Answer

Multiply each element in row $$i of$$ $$A by $$the corresponding element in column $$j of$$ $$B$$ and sum the results for each entry $$c_{ij}.$$

Matrices: Operations and Properties (Intermediate Algebra Section 3.6 Study Notes)

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