Question
If \[ A = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \] find \[ (i)\ A^2 – 5A – 14I,\quad (ii)\ A^2 – 5A + 14I,\quad (iii)\ A^3 \]
Solution
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 29 & -25 \\ -20 & 24 \end{bmatrix} \]Step 2: Compute \(A^2 – 5A\)
\[ A^2 – 5A = \begin{bmatrix} 29 & -25 \\ -20 & 24 \end{bmatrix} – \begin{bmatrix} 15 & -25 \\ -20 & 10 \end{bmatrix} = \begin{bmatrix} 14 & 0 \\ 0 & 14 \end{bmatrix} = 14I \]Step 3: Find \(A^2 – 5A – 14I\)
\[ = 14I – 14I = O \]Step 4: Find \(A^2 – 5A + 14I\)
\[ = 14I + 14I = 28I \]Step 5: Find \(A^3\)
\[ A^2 = 5A + 14I \] \[ A^3 = A \cdot A^2 = A(5A + 14I) \] \[ = 5A^2 + 14A \] Substitute \(A^2\): \[ = 5(5A + 14I) + 14A = 25A + 70I + 14A = 39A + 70I \]Final Answers
\[
A^2 – 5A – 14I = O
\]
\[
A^2 – 5A + 14I = 28I
\]
\[
A^3 = 39A + 70I
\]