If `A=[(3 ,1),(-1, 2)]` and `I=[(1, 0), (0 ,1)]` , then find `lambda` so that `A^2=5A+lambdaI`

Explanation

Given that,
`A=[(3 ,1),(-1, 2)]`

and `I=[(1 ,0),( 0 ,1)]`
`A^2=A*A`
`=[(3 ,1),(-1, 2)][(3 ,1),(-1, 2)]`
`=[(9-1,3+2),(-3-2,-1+4)]`
`=[(8,5),(-5,3)]`
Now,
`A^2=5A+lambdaI`
`=>[(8,5),(-5,3)]=5[(3 ,1),(-1, 2)]+lambda[(1 ,0),( 0 1)]`
`=>[(8,5),(-5,3)]=[(15 ,5),(-5, 10)]+[(lambda ,0),( 0 ,lambda)]`
`=>[(8,5),(-5,3)]=[(15+lambda ,5),(-5, 10+lambda)]`
The corresponding elements of two equal matrices are equal.

`⇒ 8=15+λ`

`⇒ 8−15=λ`

`∴ λ=−7`