Finding Matrix C
Question:
If \[ A=\begin{bmatrix}1 & -3 & 2 \\ 2 & 0 & 2\end{bmatrix}, \quad B=\begin{bmatrix}2 & -1 & -1 \\ 1 & 0 & -1\end{bmatrix} \] find matrix \(C\) such that: \[ A + B + C = 0 \]
If \[ A=\begin{bmatrix}1 & -3 & 2 \\ 2 & 0 & 2\end{bmatrix}, \quad B=\begin{bmatrix}2 & -1 & -1 \\ 1 & 0 & -1\end{bmatrix} \] find matrix \(C\) such that: \[ A + B + C = 0 \]
Solution:
Step 1: Rearrange the equation
\[ C = -(A + B) \]Step 2: Compute \(A + B\)
\[ A+B = \begin{bmatrix} 1+2 & -3+(-1) & 2+(-1) \\ 2+1 & 0+0 & 2+(-1) \end{bmatrix} = \begin{bmatrix} 3 & -4 & 1 \\ 3 & 0 & 1 \end{bmatrix} \]Step 3: Find \(C\)
\[ C = \begin{bmatrix} -3 & 4 & -1 \\ -3 & 0 & -1 \end{bmatrix} \]Final Answer:
\[ \boxed{ \begin{bmatrix} -3 & 4 & -1 \\ -3 & 0 & -1 \end{bmatrix} } \]