Certainly! Let's solve the given problem step by step.
### Step-by-Step Solution
Given the matrices:
[tex]\[ A = \begin{pmatrix} 2 & 3 & -4 \\ 1 & 0 & 6 \\ -2 & 1 & 5 \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} 5 & 1 & 2 \\ 6 & -1 & 4 \\ 5 & 3 & -4 \end{pmatrix} \][/tex]
We need to calculate the expression [tex]\( 2A - 3B \)[/tex].
1. Calculate [tex]\( 2A \)[/tex]:
[tex]\[ 2A = 2 \times \begin{pmatrix} 2 & 3 & -4 \\ 1 & 0 & 6 \\ -2 & 1 & 5 \end{pmatrix} \][/tex]
Multiplying each element of matrix [tex]\( A \)[/tex] by 2, we get:
[tex]\[ 2A = \begin{pmatrix} 2 \times 2 & 2 \times 3 & 2 \times -4 \\ 2 \times 1 & 2 \times 0 & 2 \times 6 \\ 2 \times -2 & 2 \times 1 & 2 \times 5 \end{pmatrix} = \begin{pmatrix} 4 & 6 & -8 \\ 2 & 0 & 12 \\ -4 & 2 & 10 \end{pmatrix} \][/tex]
2. Calculate [tex]\( 3B \)[/tex]:
[tex]\[ 3B = 3 \times \begin{pmatrix} 5 & 1 & 2 \\ 6 & -1 & 4 \\ 5 & 3 & -4 \end{pmatrix} \][/tex]
Multiplying each element of matrix [tex]\( B \)[/tex] by 3, we get:
[tex]\[ 3B = \begin{pmatrix} 3 \times 5 & 3 \times 1 & 3 \times 2 \\ 3 \times 6 & 3 \times -1 & 3 \times 4 \\ 3 \times 5 & 3 \times 3 & 3 \times -4 \end{pmatrix} = \begin{pmatrix} 15 & 3 & 6 \\ 18 & -3 & 12 \\ 15 & 9 & -12 \end{pmatrix} \][/tex]
3. Calculate [tex]\( 2A - 3B \)[/tex]:
[tex]\[ 2A - 3B = \begin{pmatrix} 4 & 6 & -8 \\ 2 & 0 & 12 \\ -4 & 2 & 10 \end{pmatrix} - \begin{pmatrix} 15 & 3 & 6 \\ 18 & -3 & 12 \\ 15 & 9 & -12 \end{pmatrix} \][/tex]
Subtracting corresponding elements of matrices [tex]\( 2A \)[/tex] and [tex]\( 3B \)[/tex], we get:
[tex]\[ 2A - 3B = \begin{pmatrix} 4 - 15 & 6 - 3 & -8 - 6 \\ 2 - 18 & 0 - (-3) & 12 - 12 \\ -4 - 15 & 2 - 9 & 10 - (-12) \end{pmatrix} \][/tex]
[tex]\[ 2A - 3B = \begin{pmatrix} -11 & 3 & -14 \\ -16 & 3 & 0 \\ -19 & -7 & 22 \end{pmatrix} \][/tex]
So, the final result of [tex]\( 2A - 3B \)[/tex] is:
[tex]\[ \begin{pmatrix} -11 & 3 & -14 \\ -16 & 3 & 0 \\ -19 & -7 & 22 \end{pmatrix} \][/tex]
This completes our step-by-step solution.
