If A = find A² – 5A + 4I and hence find a matrix X such that A² – 5A + 4I + X = 0.

OR

If A = , find (A’)^-1

As, A =

∴ A² =

Using matrix multiplication method we can write it as-

A² =

⇒ A² =

∴ A² – 5A + 4I =

⇒ A² – 5A + 4I =

⇒ A² – 5A + 4I =

⇒ A² – 5A + 4I =

Now we need to find X such that -

A² – 5A + 4I + X = 0

⇒

∴ X =

⇒ X =

OR

As we know that transpose of a matrix is given by interchanging rows with respective columns.

∴ A’ = A^T =

Inverse of any matrix(say B) is given by:

B^-1 =

Assume, B =

Determinant of B = |B| =

Expanding about first row-

|B| = 1(-1-8)-0(-2-6)-2(-8+3)

⇒ |B| = -9+10 = 1

Adjoint of a matrix is given by the transpose of cofactor matrix.

Co-factor matrix of B =

∴ adj(B) =

∴ (A’)^-1 =

Hence,

(A’)^-1 =