Questions: [ left[beginarraycc 5 -7 4 6 endarrayright]left[beginarrayc 4 -8 endarrayright] ]

Transcript text: \[ \left[\begin{array}{cc} 5 & -7 \\ 4 & 6 \end{array}\right]\left[\begin{array}{c} 4 \\ -8 \end{array}\right] \]

Solution

Solution Steps

To find the product of a matrix and a vector, we perform matrix multiplication. This involves taking the dot product of each row of the matrix with the vector. For a matrix $A$ with dimensions $m \times n$ and a vector $v$ with dimensions $n \times 1$ , the resulting product will be a vector with dimensions $m \times 1$ .

Step 1: Define the Matrix and Vector

We are given a matrix $A$ and a vector $v$ . The matrix $A$ is: $A = \begin{bmatrix} 5 & -7 \\ 4 & 6 \end{bmatrix}$ and the vector $v$ is: $v = \begin{bmatrix} 4 \\ -8 \end{bmatrix}$

Step 2: Perform Matrix-Vector Multiplication

To find the product $Av$ , we perform the matrix-vector multiplication. This involves taking the dot product of each row of the matrix with the vector.

For the first row of $A$ : $5 \times 4 + (-7) \times (-8) = 20 + 56 = 76$
For the second row of $A$ : $4 \times 4 + 6 \times (-8) = 16 - 48 = -32$