The matrices M and N are defined as follows: M = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmat… — ACT

The matrices $M$ and $N$ are defined as follows:

M = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}, \quad N = \begin{bmatrix} 2 & 5 \\ 0 & -1 \end{bmatrix}

If the product matrix $P$ is defined by the equation $P = MN$ , what is the value of the element in the first row and second column of $P$ ?

Answer: 22

The value of the element in the first row and second column of

P

is 22.

To find the element in the first row and second column of the product matrix

P = MN

, we calculate the dot product of the first row of matrix

M

(

[4, -2]

) and the second column of matrix

N

(

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

4(5) + (-2)(-1) = 20 + 2 = 22

Determine the row of the left matrix and the column of the right matrix needed for the target element.

To find the element in row 1, column 2 of

P = MN

, use the first row of

M

, which is

[4, -2]

, and the second column of

N

, which is

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

The entry in row

i

and column

j

of a product matrix is the dot product of row

i

of the first matrix and column

j

of the second matrix.

Compute the dot product of the selected row and column.

(4 \times 5) + (-2 \times -1) = 20 + 2

Multiply corresponding elements and sum the products.

Simplify to get the final numeric answer.

Adding 20 and 2 yields 22.

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix.

Question