The matrices A and B are defined as follows: A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatr… — ACT

The matrices $A$ and $B$ are defined as follows:

A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}

What is the product matrix $AB$ ?

$\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}$ Answer
B
$\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}$
C
$\begin{bmatrix} 7 & 3 \\ 2 & 6 \end{bmatrix}$
D
$\begin{bmatrix} 5 & 1 \\ 1 & 5 \end{bmatrix}$
E
$\begin{bmatrix} 6 & 0 \\ 6 & 6 \end{bmatrix}$

Answer

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

The correct matrix is obtained by performing row-by-column matrix multiplication, which yields row 1 entries of 6 and 2, and row 2 entries of 3 and 7.

Step-by-Step Solution

Multiply the first row of

A

by the first column of

B

to find the element in the first row, first column of the product matrix.

2(3) + 0(0) = 6

The row-by-column rule dictates that the entry at row

i

, column

j

of the product is the dot product of row

i

of the first matrix and column

j

of the second matrix.

Multiply the first row of

A

by the second column of

B

to find the element in the first row, second column.

2(1) + 0(2) = 2

This determines the top-right entry of the resulting matrix.

Multiply the second row of

A

by the first column of

B

to find the element in the second row, first column.

1(3) + 3(0) = 3

This determines the bottom-left entry of the resulting matrix.

Multiply the second row of

A

by the second column of

B

to find the element in the second row, second column.

1(1) + 3(2) = 1 + 6 = 7

This determines the bottom-right entry of the resulting matrix.

Key Concept

Matrix multiplication involves computing the dot product of the rows of the first matrix with the columns of the second matrix.

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