Finding Matrix C
Question:
If \[ A=\begin{bmatrix}9 & 1 \\ 7 & 8\end{bmatrix}, \quad B=\begin{bmatrix}1 & 5 \\ 7 & 12\end{bmatrix} \] find matrix \(C\) such that: \[ 5A + 3B + 2C = 0 \]
If \[ A=\begin{bmatrix}9 & 1 \\ 7 & 8\end{bmatrix}, \quad B=\begin{bmatrix}1 & 5 \\ 7 & 12\end{bmatrix} \] find matrix \(C\) such that: \[ 5A + 3B + 2C = 0 \]
Solution:
Step 1: Rearrange the equation
\[ 2C = -(5A + 3B) \]Step 2: Compute \(5A\)
\[ 5A = \begin{bmatrix} 45 & 5 \\ 35 & 40 \end{bmatrix} \]Step 3: Compute \(3B\)
\[ 3B = \begin{bmatrix} 3 & 15 \\ 21 & 36 \end{bmatrix} \]Step 4: Add \(5A + 3B\)
\[ = \begin{bmatrix} 45+3 & 5+15 \\ 35+21 & 40+36 \end{bmatrix} = \begin{bmatrix} 48 & 20 \\ 56 & 76 \end{bmatrix} \]Step 5: Find \(2C\)
\[ 2C = \begin{bmatrix} -48 & -20 \\ -56 & -76 \end{bmatrix} \]Step 6: Divide by 2
\[ C = \begin{bmatrix} -24 & -10 \\ -28 & -38 \end{bmatrix} \]Final Answer:
\[ \boxed{ \begin{bmatrix} -24 & -10 \\ -28 & -38 \end{bmatrix} } \]