Question

i) lf the inverse of matrix $A$ exist. Then matrix is non singular.
ii) lf a square, non-singular matrix $A$ statisfies $A^2-A+2I=0$ then $A^{-1}=\frac{1}{2}(I+A)$. Which of the following statement is correct:

• A. Only $i$
• B. Only $ii$
• C. Both $i$ and $ii$
• D. Neither $i$ nor $ii$

Answer 1

The correct option is A Only $i$

Inverse of a matrix exists if and only if the matrix is non-singular or $det\left(A\right)\ne 0$
Now if
$A^2-A+2I=0$ where A is a non-singular matrix
Multiplying by $A^{-1}$ on both sides give us
$A^{-1}\left(A^2\right)-A^{-1}A+2A^{-1}=0$
$A-I+2A^{-1}=0$
$2A^{-1}=I-A$
$A^{-1}=\frac{I-A}{2}$