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 $AB$ ?

$\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}$ Cevap
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
$\begin{bmatrix} 6 & 0 \\ 6 & 6 \end{bmatrix}$

Cevap

\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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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