Let A = [2 3; 5] and B = [0 4; -1 6]; find 4AB.

Steps to Solve

1. Identify the matrices

Given matrices are:

$$ A = \begin{pmatrix} 2 & 3 \ 5 & 0 \end{pmatrix} $$

$$ B = \begin{pmatrix} 0 & 4 \ -1 & 6 \end{pmatrix} $$

2. Calculate the matrix product AB

The product of matrices $A$ and $B$ is calculated using the formula for matrix multiplication, where each element of the resulting matrix is the sum of the products of the corresponding entries.

Calculating the first row, first column:

$$ (20) + (3-1) = 0 - 3 = -3 $$

Calculating the first row, second column:

$$ (24) + (36) = 8 + 18 = 26 $$

Calculating the second row, first column:

$$ (50) + (0-1) = 0 + 0 = 0 $$

Calculating the second row, second column:

$$ (54) + (06) = 20 + 0 = 20 $$

Thus, the product is:

$$ AB = \begin{pmatrix} -3 & 26 \ 0 & 20 \end{pmatrix} $$

3. Scale the product by 4

Now, we multiply the resulting matrix $AB$ by 4:

$$ 4AB = 4 \cdot \begin{pmatrix} -3 & 26 \ 0 & 20 \end{pmatrix} $$

This results in:

$$ 4AB = \begin{pmatrix} 4 \cdot -3 & 4 \cdot 26 \ 4 \cdot 0 & 4 \cdot 20 \end{pmatrix} $$

Calculating each element:

• First row, first column: $4 \cdot -3 = -12$
• First row, second column: $4 \cdot 26 = 104$
• Second row, first column: $4 \cdot 0 = 0$
• Second row, second column: $4 \cdot 20 = 80$

Thus,

$$ 4AB = \begin{pmatrix} -12 & 104 \ 0 & 80 \end{pmatrix} $$