TIFR-2019-Maths-B: 2

Question

True/False Question :

If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.

Answer 1

$\text{Matrix A belongs to a set of real matrices of size $10 \times 10.$}$

$\text{Now, $A^2 + A + I = 0$}$

$\text{$\Rightarrow A(A+I) + I = 0$}$

$\text{$\Rightarrow A(-(A+I)) = I$ }$

$\text{$\Rightarrow \det(A(-(A+I))) = \det(I)$ }$

$\text{$\Rightarrow$ det(A) $\times$ det(-(A+I)) = 1}$

$\text{ $\Rightarrow \det(A) \neq 0$}$

$\text{So, matrix A is invertible}$