Question

g Problem 2. (1.4 points) Suppose A is an n × n matrix, such that A 2 + 3A − 4I = 0 where I and 0 are the n × n identity and zero matrix respectively. What can you say about the invertibility of matrix A?

Answer 1

Answer:

The correct answer is $A^{-1}$ = $\frac{1}{4}$ A + $\frac{3}{4 }$ I

Step-by-step explanation:

Since A is an n × n invertible non singular matrix, $A^{-1}$ exists.

Given equation is $A^{2}$ + 3A -4I = O where I and O are the n × n identity and zero matrix respectively.

⇒ $A^{2}$ + 3A -4I = O

⇒ $A^{-1}$ ( $A^{2}$ + 3A -4I ) = $A^{-1}$ × O

⇒ $A^{-1}$ AA + 3 $A^{-1}$A - 4$A^{-1}$I = O

⇒ A + 3I - 4$A^{-1}$ = O

⇒ A + 3I = 4 $A^{-1}$

⇒ $A^{-1}$ = $\frac{1}{4}$ A + $\frac{3}{4 }$ I

Thus the value of  $A^{-1}$ can be given by the above equation.