Matrix and inverse matrix question?

[nishantve1]

Homework Statement

how do I solve this question
https://www.dropbox.com/s/bn958xnm5483s6u/photo%20%281%29.JPG
(This link if the image is not visible : https://www.dropbox.com/s/bn958xnm5483s6u/photo (1).JPG)
just so the equation is not clear it says A² = λA - 2I

The inverse should be found through the equation in the question and not through the adjoint method

Homework Equations

The Attempt at a Solution

The equation says
A² = λA - 2I
So, I mulitplied by A^-1
This gives
A² A^-1 = λA A^-1 - 2IA^-1
{A^-1 . A = I}

now the equation becomes
A = λI - 2A^-1

I am stuck here since I cannot simply find A^-1 by the adjoint method this equation has two unknowns A^-1 and λ . Or maybe I am misinterpreting the question IDK

 

[micromass]

But $\lambda$ is not an unkown.

You are given a very specific matrix A. From your specific matrix A, and from your equation $A^2= \lambda A- 2I$, you can deduce what $\lambda$ is.

 

[nishantve1]

How did I miss that . Thanks

 

[HallsofIvy]

I see no point in looking for specific values of $\lambda$. To find the multiplicative inverse of A:
1) algebraically manipulate the equation, $A^2= \lambda A- 2I$, to get "I" alone on the right.
2) factor out an "A".