Question

consider the matrix equation below.
\

$$\begin{bmatrix} 8 & -9 \\\\ -4 & 6 \\end{bmatrix}$$

\cdot \

$$\begin{bmatrix} 5 \\\\ 7 \\end{bmatrix}$$

= \

$$\begin{bmatrix} a \\\\ b \\end{bmatrix}$$

what are the values of a and b?
note: your answers should be integers.
a = \boxed{}

b = \boxed{}

Explanation:

Step1: Calculate A (first element)

To find \( A \), we perform the matrix multiplication for the first row of the matrix and the column vector. The formula for the product of a \( 2\times2 \) matrix \(

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

\) and a \( 2\times1 \) vector \(

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

\) is \(

$$\begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$$

\). For the first element (A), we use the first row: \( 8\times5 + (-9)\times7 \).

$$ 8\times5 + (-9)\times7 = 40 - 63 = -23 $$

Step2: Calculate B (second element)

For the second element (B), we use the second row of the matrix: \( -4\times5 + 6\times7 \).

$$ -4\times5 + 6\times7 = -20 + 42 = 22 $$

Answer:

\( A = -23 \)
\( B = 22 \)