Question

If $|A|\ne 0$ and $(A-2I)(A-3I)=0$ then $A^{-1}=?$

• A. $\frac{A-5I}{6}$
• B. $\frac{5I-A}{5}$
• C. $\frac{5A-I}{6}$
• D. $\frac{5I-A}{6}$

Answer 1

Answer: (D)

$\frac{5I-A}{6}$

$\begin{array}{c}(A-2I)(A-3I)=0,A^{-1}=?\\ \Rightarrow A\cdot A-3A\cdot I-2I\cdot A+6I\cdot I=O\left[\because A\cdot I=A\right]\\ \Rightarrow A\cdot A-3A-2A+6I=0\\ \Rightarrow A\cdot A+6I=5A\end{array}$

Multiply $A^{-1}$ on both side

$\begin{array}{c}\Rightarrow A\cdot A\left(A^{-1}\right)+6I\left(A^{-1}\right)=5A\left(A^{-1}\right)\\ \Rightarrow A\cdot \left(A{A}^{-1}\right)+6A^{-1}=5I\left[\because A{A}^{-1}=I\&\because I{A}^{-1}=A^{-1}\right]\\ \Rightarrow AI+6A^{-1}=5I\\ \Rightarrow A^{-1}=\frac{5I-AI}{6}\\ \Rightarrow A^{-1}=\frac{5I-A}{6}\end{array}$