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

Steps to Solve

1. Identify the Matrices

The matrices are defined as follows:

$$ A = \begin{bmatrix} 1 & 2 & 3 \ 5 \end{bmatrix} $$

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

2. Matrix Multiplication

For matrix multiplication $AB$, we need to ensure that the number of columns in $A$ matches the number of rows in $B$.

Here, matrix A has dimensions $2 \times 3$ and matrix B has dimensions $3 \times 2$.

The result of $AB$ will be a $2 \times 2$ matrix.

To calculate the elements of $AB$, we perform the following calculations:

• First element:

$$ C_{11} = 1 \cdot 0 + 2 \cdot 4 + 3 \cdot 6 = 0 + 8 + 18 = 26 $$

• Second element:

$$ C_{12} = 1 \cdot -1 + 2 \cdot 6 + 3 \cdot 0 = -1 + 12 + 0 = 11 $$

• Third element:

$$ C_{21} = 5 \cdot 0 + 0 \cdot 4 + 0 \cdot 6 = 0 + 0 + 0 = 0 $$

• Fourth element:

$$ C_{22} = 5 \cdot -1 + 0 \cdot 6 + 0 \cdot 0 = -5 + 0 + 0 = -5 $$

So, we have

$$ AB = \begin{bmatrix} 26 & 11 \ 0 & -5 \end{bmatrix} $$

3. Scalar Multiplication

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

$$ 4AB = 4 \cdot \begin{bmatrix} 26 & 11 \ 0 & -5 \end{bmatrix} = \begin{bmatrix} 4 \cdot 26 & 4 \cdot 11 \ 4 \cdot 0 & 4 \cdot -5 \end{bmatrix} $$

Calculating the elements results in:

• First element:

$$ 4 \cdot 26 = 104 $$

• Second element:

$$ 4 \cdot 11 = 44 $$

• Third element:

$$ 4 \cdot 0 = 0 $$

• Fourth element:

$$ 4 \cdot -5 = -20 $$

Thus,

$$ 4AB = \begin{bmatrix} 104 & 44 \ 0 & -20 \end{bmatrix} $$