1. Find the adjoint of the matrix  

The co factor of a is d, the co factor of b is -c, the co factor of c is -b and the co factor of d is a.  The matrix formed by the co factors taken in order is the co factor matrix of A.

  The co factor matrix of A is =   Taking transpose of the co factor matrix, we get the adjoint  of A.

        The adjoint of  

 2. State and Prove Reversal Law for Inverses .

Statement:

       If A,B are any two non-singular matrices of the same order, then AB is also non-singular and

       (AB)^-1 = B^-1A^-1 ie, the inverse of a  product is product of  inverses taken in the reverse order.

Proof

3. Prove that "For any non-singular matrix A, (A^T)^-1 = (A^-1)^T.

 Proof

  Taking transpose on both sides of AA^-1 = I

  We have (AA^-1)^T = I^T

  By reversal law for transposes, we get

         (A^-1)^T A^T = I  → (1)

 Similarly by taking transpose on both of A^-1A = I, we have

         (A^T) (A^-1)^T = I → (2)

  From (1) and (2)

        (A^-1)^T A^T = A^T (A^-1)^T = I

        (A^-1)^T is the inverse of A^T

A is a non-singular matrix.  Hence it is invertible.  The matrix formed by the co factors is

7. Solve by matrix inversion method x + y = 3, 2x + 3y = 8.

 The given system of equations can be written in the form of

  The highest order minor of A is given by   .  Since the second order minor vanishes .  We have to try for at least one non-zero first order minor, ie, atleast one non-zero element of A.  This is possible because A has non-zero elements.

    .

10. Find the rank of the matrix .