Finding Matrix X
Question:
If \[ 2A + B + X = 0 \] where \[ A=\begin{bmatrix}-1 & 2 \\ 3 & 4\end{bmatrix}, \quad B=\begin{bmatrix}3 & -2 \\ 1 & 5\end{bmatrix} \] find matrix \(X\).
If \[ 2A + B + X = 0 \] where \[ A=\begin{bmatrix}-1 & 2 \\ 3 & 4\end{bmatrix}, \quad B=\begin{bmatrix}3 & -2 \\ 1 & 5\end{bmatrix} \] find matrix \(X\).
Solution:
Step 1: Rearrange the equation
\[ X = -(2A + B) \]Step 2: Compute \(2A\)
\[ 2A = \begin{bmatrix} -2 & 4 \\ 6 & 8 \end{bmatrix} \]Step 3: Add \(2A + B\)
\[ = \begin{bmatrix} -2+3 & 4+(-2) \\ 6+1 & 8+5 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 7 & 13 \end{bmatrix} \]Step 4: Find \(X\)
\[ X = \begin{bmatrix} -1 & -2 \\ -7 & -13 \end{bmatrix} \]Final Answer:
\[ \boxed{ \begin{bmatrix} -1 & -2 \\ -7 & -13 \end{bmatrix} } \]