Finding Matrix X
Question:
If \[ A=\begin{bmatrix}2 & -2 \\ 4 & 2 \\ -5 & 1\end{bmatrix}, \quad B=\begin{bmatrix}8 & 0 \\ 4 & -2 \\ 3 & 6\end{bmatrix} \] find matrix \(X\) such that: \[ 2A + 3X = 5B \]
If \[ A=\begin{bmatrix}2 & -2 \\ 4 & 2 \\ -5 & 1\end{bmatrix}, \quad B=\begin{bmatrix}8 & 0 \\ 4 & -2 \\ 3 & 6\end{bmatrix} \] find matrix \(X\) such that: \[ 2A + 3X = 5B \]
Solution:
Step 1: Rearrange the equation
\[ 3X = 5B – 2A \]Step 2: Compute \(5B\)
\[ 5B = \begin{bmatrix} 40 & 0 \\ 20 & -10 \\ 15 & 30 \end{bmatrix} \]Step 3: Compute \(2A\)
\[ 2A = \begin{bmatrix} 4 & -4 \\ 8 & 4 \\ -10 & 2 \end{bmatrix} \]Step 4: Compute \(5B – 2A\)
\[ = \begin{bmatrix} 40-4 & 0-(-4) \\ 20-8 & -10-4 \\ 15-(-10) & 30-2 \end{bmatrix} = \begin{bmatrix} 36 & 4 \\ 12 & -14 \\ 25 & 28 \end{bmatrix} \]Step 5: Divide by 3
\[ X = \frac{1}{3} \begin{bmatrix} 36 & 4 \\ 12 & -14 \\ 25 & 28 \end{bmatrix} = \begin{bmatrix} 12 & \frac{4}{3} \\ 4 & -\frac{14}{3} \\ \frac{25}{3} & \frac{28}{3} \end{bmatrix} \]Final Answer:
\[ \boxed{ \begin{bmatrix} 12 & \frac{4}{3} \\ 4 & -\frac{14}{3} \\ \frac{25}{3} & \frac{28}{3} \end{bmatrix} } \]