Question-53
\(\displaystyle A\) is a square matrix that satisfies the following equation:
\[ \begin{equation*} A^{3} -4A^{2} +3A-5I=0 \end{equation*} \]
where \(\displaystyle I\) is the identity matrix. Which of the following statements are true?
Factorize the matrix expression on the LHS after moving \(5I\) out of the way.
We can work with the matrix equation on the left as though we were factorizing a polynomial:
\[ \begin{equation*} \begin{aligned} A^{3} -4A^{2} +3A-5I & =0\\ A^{3} -4A^{2} +3A & =5I\\ A\left( A^{2} -4A+3I\right) & =5I\\ A\left(\frac{A^{2} -4A+3I}{5}\right) & =I \end{aligned} \end{equation*} \]
Since \(\displaystyle A\) is a square matrix and since we have \(\displaystyle AB=I\) where \(\displaystyle B\) is some other square matrix, \(\displaystyle A\) is invertible and \(\displaystyle B\) is its inverse. We can factor the inverse as follows:
\[ \begin{equation*} A^{-1} =\frac{1}{5}( A-3I)( A-I) \end{equation*} \]