To find the matrix [tex]\(2A - 2B\)[/tex], let's break down the solution step-by-step.
First, we need to calculate [tex]\(2A\)[/tex] and [tex]\(2B\)[/tex], and then subtract [tex]\(2B\)[/tex] from [tex]\(2A\)[/tex].
Given:
[tex]\[ A = \left[\begin{array}{rr}2 & 0 \\ -3 & 1\end{array}\right] \][/tex]
[tex]\[ B = \left[\begin{array}{rr}0 & 1 \\ -2 & 3\end{array}\right] \][/tex]
### Step 1: Calculate [tex]\(2A\)[/tex]
To find [tex]\(2A\)[/tex], multiply every element of the matrix [tex]\(A\)[/tex] by 2:
[tex]\[ 2A = 2 \times \left[\begin{array}{rr}2 & 0 \\ -3 & 1\end{array}\right] \][/tex]
[tex]\[ 2A = \left[\begin{array}{rr}2 \times 2 & 2 \times 0 \\ 2 \times -3 & 2 \times 1\end{array}\right] \][/tex]
[tex]\[ 2A = \left[\begin{array}{rr}4 & 0 \\ -6 & 2\end{array}\right] \][/tex]
### Step 2: Calculate [tex]\(2B\)[/tex]
To find [tex]\(2B\)[/tex], multiply every element of the matrix [tex]\(B\)[/tex] by 2:
[tex]\[ 2B = 2 \times \left[\begin{array}{rr}0 & 1 \\ -2 & 3\end{array}\right] \][/tex]
[tex]\[ 2B = \left[\begin{array}{rr}2 \times 0 & 2 \times 1 \\ 2 \times -2 & 2 \times 3\end{array}\right] \][/tex]
[tex]\[ 2B = \left[\begin{array}{rr}0 & 2 \\ -4 & 6\end{array}\right] \][/tex]
### Step 3: Calculate [tex]\(2A - 2B\)[/tex]
Now subtract matrix [tex]\(2B\)[/tex] from matrix [tex]\(2A\)[/tex]:
[tex]\[ 2A - 2B = \left[\begin{array}{rr}4 & 0 \\ -6 & 2\end{array}\right] - \left[\begin{array}{rr}0 & 2 \\ -4 & 6\end{array}\right] \][/tex]
Perform the subtraction for each element:
[tex]\[ \left[\begin{array}{rr}4 - 0 & 0 - 2 \\ -6 - (-4) & 2 - 6\end{array}\right] \][/tex]
[tex]\[ 2A - 2B = \left[\begin{array}{rr}4 & -2 \\ -2 & -4\end{array}\right] \][/tex]
Thus, the matrix [tex]\(2A - 2B\)[/tex] is:
[tex]\[ \left[\begin{array}{rr}4 & -2 \\ -2 & -4\end{array}\right] \][/tex]
