Answer :
To solve the matrix expression \( 3[C] + 2[A] \), we need to perform the following steps:
1. Calculate \( 3[C] \) by multiplying each element of matrix \( C \) by 3.
2. Calculate \( 2[A] \) by multiplying each element of matrix \( A \) by 2.
3. Add the resulting matrices from steps 1 and 2 element-wise.
### Step 1: Calculate \( 3[C] \)
Matrix \( C \):
[tex]\[ \begin{array}{rrrrr} -6 & 2 & -6 & -1 & 0 \\ -10 & 9 & -4 & 9 & 5 \\ 7 & -7 & 6 & 6 & -9 \\ -5 & 8 & 8 & 10 & -3 \\ 0 & 4 & 1 & 7 & -5 \end{array} \][/tex]
Multiplying each element by 3:
[tex]\[ 3[C] = \begin{array}{rrrrr} -18 & 6 & -18 & -3 & 0 \\ -30 & 27 & -12 & 27 & 15 \\ 21 & -21 & 18 & 18 & -27 \\ -15 & 24 & 24 & 30 & -9 \\ 0 & 12 & 3 & 21 & -15 \end{array} \][/tex]
### Step 2: Calculate \( 2[A] \)
Matrix \( A \):
[tex]\[ \begin{array}{rrrrr} -3 & -7 & 1 & -7 & 4 \\ 2 & -9 & 8 & 7 & -1 \\ 0 & -8 & 0 & 3 & -3 \\ -5 & -4 & 7 & 5 & -10 \\ 9 & 1 & -4 & 2 & 4 \end{array} \][/tex]
Multiplying each element by 2:
[tex]\[ 2[A] = \begin{array}{rrrrr} -6 & -14 & 2 & -14 & 8 \\ 4 & -18 & 16 & 14 & -2 \\ 0 & -16 & 0 & 6 & -6 \\ -10 & -8 & 14 & 10 & -20 \\ 18 & 2 & -8 & 4 & 8 \end{array} \][/tex]
### Step 3: Calcluate \( 3[C] + 2[A] \)
Adding the matrices \( 3[C] \) and \( 2[A] \) element-wise:
[tex]\[ \begin{array}{rrrrr} -18 + (-6) & 6 + (-14) & -18 + 2 & -3 + (-14) & 0 + 8 \\ -30 + 4 & 27 + (-18) & -12 + 16 & 27 + 14 & 15 + (-2) \\ 21 + 0 & -21 + (-16) & 18 + 0 & 18 + 6 & -27 + (-6) \\ -15 + (-10) & 24 + (-8) & 24 + 14 & 30 + 10 & -9 + (-20) \\ 0 + 18 & 12 + 2 & 3 + (-8) & 21 + 4 & -15 + 8 \end{array} \][/tex]
This results in:
[tex]\[ \begin{array}{rrrrr} -24 & -8 & -16 & -17 & 8 \\ -26 & 9 & 4 & 41 & 13 \\ 21 & -37 & 18 & 24 & -33 \\ -25 & 16 & 38 & 40 & -29 \\ 18 & 14 & -5 & 25 & -7 \end{array} \][/tex]
Therefore, the solution to the matrix expression \( 3[C] + 2[A] \) is:
[tex]\[ \begin{array}{rrrrr} -24 & -8 & -16 & -17 & 8 \\ -26 & 9 & 4 & 41 & 13 \\ 21 & -37 & 18 & 24 & -33 \\ -25 & 16 & 38 & 40 & -29 \\ 18 & 14 & -5 & 25 & -7 \end{array} \][/tex]
1. Calculate \( 3[C] \) by multiplying each element of matrix \( C \) by 3.
2. Calculate \( 2[A] \) by multiplying each element of matrix \( A \) by 2.
3. Add the resulting matrices from steps 1 and 2 element-wise.
### Step 1: Calculate \( 3[C] \)
Matrix \( C \):
[tex]\[ \begin{array}{rrrrr} -6 & 2 & -6 & -1 & 0 \\ -10 & 9 & -4 & 9 & 5 \\ 7 & -7 & 6 & 6 & -9 \\ -5 & 8 & 8 & 10 & -3 \\ 0 & 4 & 1 & 7 & -5 \end{array} \][/tex]
Multiplying each element by 3:
[tex]\[ 3[C] = \begin{array}{rrrrr} -18 & 6 & -18 & -3 & 0 \\ -30 & 27 & -12 & 27 & 15 \\ 21 & -21 & 18 & 18 & -27 \\ -15 & 24 & 24 & 30 & -9 \\ 0 & 12 & 3 & 21 & -15 \end{array} \][/tex]
### Step 2: Calculate \( 2[A] \)
Matrix \( A \):
[tex]\[ \begin{array}{rrrrr} -3 & -7 & 1 & -7 & 4 \\ 2 & -9 & 8 & 7 & -1 \\ 0 & -8 & 0 & 3 & -3 \\ -5 & -4 & 7 & 5 & -10 \\ 9 & 1 & -4 & 2 & 4 \end{array} \][/tex]
Multiplying each element by 2:
[tex]\[ 2[A] = \begin{array}{rrrrr} -6 & -14 & 2 & -14 & 8 \\ 4 & -18 & 16 & 14 & -2 \\ 0 & -16 & 0 & 6 & -6 \\ -10 & -8 & 14 & 10 & -20 \\ 18 & 2 & -8 & 4 & 8 \end{array} \][/tex]
### Step 3: Calcluate \( 3[C] + 2[A] \)
Adding the matrices \( 3[C] \) and \( 2[A] \) element-wise:
[tex]\[ \begin{array}{rrrrr} -18 + (-6) & 6 + (-14) & -18 + 2 & -3 + (-14) & 0 + 8 \\ -30 + 4 & 27 + (-18) & -12 + 16 & 27 + 14 & 15 + (-2) \\ 21 + 0 & -21 + (-16) & 18 + 0 & 18 + 6 & -27 + (-6) \\ -15 + (-10) & 24 + (-8) & 24 + 14 & 30 + 10 & -9 + (-20) \\ 0 + 18 & 12 + 2 & 3 + (-8) & 21 + 4 & -15 + 8 \end{array} \][/tex]
This results in:
[tex]\[ \begin{array}{rrrrr} -24 & -8 & -16 & -17 & 8 \\ -26 & 9 & 4 & 41 & 13 \\ 21 & -37 & 18 & 24 & -33 \\ -25 & 16 & 38 & 40 & -29 \\ 18 & 14 & -5 & 25 & -7 \end{array} \][/tex]
Therefore, the solution to the matrix expression \( 3[C] + 2[A] \) is:
[tex]\[ \begin{array}{rrrrr} -24 & -8 & -16 & -17 & 8 \\ -26 & 9 & 4 & 41 & 13 \\ 21 & -37 & 18 & 24 & -33 \\ -25 & 16 & 38 & 40 & -29 \\ 18 & 14 & -5 & 25 & -7 \end{array} \][/tex]
