Certainly! Let's solve the given problem step by step. We are given two matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] and we need to find the matrix [tex]\( 2A - 2B \)[/tex].
The matrices are as follows:
[tex]\[ A = \begin{bmatrix} 2 & 0 \\ -3 & 1 \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \][/tex]
First, we need to compute the matrix [tex]\( 2A \)[/tex]. This involves multiplying every element of matrix [tex]\( A \)[/tex] by 2:
[tex]\[ 2A = 2 \cdot \begin{bmatrix} 2 & 0 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 2 \cdot 2 & 2 \cdot 0 \\ 2 \cdot (-3) & 2 \cdot 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ -6 & 2 \end{bmatrix} \][/tex]
Next, we need to compute the matrix [tex]\( 2B \)[/tex]. This involves multiplying every element of matrix [tex]\( B \)[/tex] by 2:
[tex]\[ 2B = 2 \cdot \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} 2 \cdot 0 & 2 \cdot 1 \\ 2 \cdot (-2) & 2 \cdot 3 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ -4 & 6 \end{bmatrix} \][/tex]
Now, we need to find the matrix [tex]\( 2A - 2B \)[/tex]. This involves subtracting the corresponding elements of [tex]\( 2B \)[/tex] from [tex]\( 2A \)[/tex]:
[tex]\[ 2A - 2B = \begin{bmatrix} 4 & 0 \\ -6 & 2 \end{bmatrix} - \begin{bmatrix} 0 & 2 \\ -4 & 6 \end{bmatrix} = \begin{bmatrix} 4 - 0 & 0 - 2 \\ -6 - (-4) & 2 - 6 \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ -2 & -4 \end{bmatrix} \][/tex]
So, the result of [tex]\( 2A - 2B \)[/tex] is:
[tex]\[ 2A - 2B = \begin{bmatrix} 4 & -2 \\ -2 & -4 \end{bmatrix} \][/tex]
This is the final answer.
