Let \( A=\left[\begin{array}{rr}3 & -2 \\ 8 & 3\end{array}\right] \) and \( B=\left[\begin{array}{rr}4 & -9 \\ -4 & -7\end{array}\right] \). Find a matrix \( X \) satisfying the given equation. \( 5 X=-2 A+3 B \) \( X=\square \) (Simplify your answer.)

To find matrix \( X \), we first substitute the given matrices \( A \) and \( B \) into the equation \( 5X = -2A + 3B \). Calculating \( -2A \): \[ -2A = -2 \left[\begin{array}{rr}3 & -2 \\ 8 & 3\end{array}\right] = \left[\begin{array}{rr}-6 & 4 \\ -16 & -6\end{array}\right] \] Calculating \( 3B \): \[ 3B = 3 \left[\begin{array}{rr}4 & -9 \\ -4 & -7\end{array}\right] = \left[\begin{array}{rr}12 & -27 \\ -12 & -21\end{array}\right] \] Now, we add \( -2A \) and \( 3B \): \[ -2A + 3B = \left[\begin{array}{rr}-6 & 4 \\ -16 & -6\end{array}\right] + \left[\begin{array}{rr}12 & -27 \\ -12 & -21\end{array}\right] \] Calculating the sum element-wise: \[ \left[\begin{array}{rr}-6 + 12 & 4 - 27 \\ -16 - 12 & -6 - 21\end{array}\right] = \left[\begin{array}{rr}6 & -23 \\ -28 & -27\end{array}\right] \] Now, we have: \[ 5X = \left[\begin{array}{rr}6 & -23 \\ -28 & -27\end{array}\right] \] To find \( X \), we divide both sides by 5: \[ X = \frac{1}{5} \left[\begin{array}{rr}6 & -23 \\ -28 & -27\end{array}\right] = \left[\begin{array}{rr}\frac{6}{5} & -\frac{23}{5} \\ -\frac{28}{5} & -\frac{27}{5}\end{array}\right] \] Thus, the simplified answer for \( X \) is: \[ X = \left[\begin{array}{rr}\frac{6}{5} & -\frac{23}{5} \\ -\frac{28}{5} & -\frac{27}{5}\end{array}\right] \]