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Find the adjoint of the given matrix and verify in each case that A. (adj A) = (adj A) =m |A|.I.

If A = , show that adj A = A.

If A = , show that adj A = 3A’.

Find the inverse of each of the matrices given below.

, when (ab – bc) 0

If A = , show that A^{-1} = A.

If A = , show that A^{-1} = A^{2}.

If A = , prove that A^{-1} = A^{3}.

If A = show that A^{-1}= A’.

Let D = diag [d_{1}, d_{2}, d_{3}], where none of d_{1}, d_{2}, d_{3} is 0; prove that D^{-1} = diag [d_{1}^{-1}, d_{2}^{-1}, d_{3}^{-1}].

If A = and B = , verify that (AB)^{-1} = B^{-1} A^{-1}.

Compute (AB)^{-1} when A = and B^{-1} -= .

Obtain the inverses of the matrices and . And, hence find the inverse of the matrix .

If A = , verify that A^{2} – 4A – I = O, and hence find A^{-1}.

Show that the matrix A = satisfies the equation

If A = , show that A^{2} + 3A + 4I_{2} = O and hence find A^{-1}.

If A = , find

If A = . Find the value of λ so that A^{2} = λA – 2I. Hence, find A^{-1}.

[CBSE 2007]

Show that the A = satisfies the equation A^{3} – A^{2} – 3A – I = O, and hence find A^{-1}.

Prove that: (i) adj I = I (ii) adj O = O (iii) I^{-1} = I.