#### Topics

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, (AT)-1 = (A-1)T.

Proof

Taking transpose on both sides of AA-1 = I

We have (AA-1)T = IT

By reversal law for transposes, we get

(A-1)T AT = I  → (1)

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

(AT) (A-1)T = I → (2)

From (1) and (2)

(A-1)T AT = AT (A-1)T = I (A-1)T is the inverse of AT 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 . Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only!
Paid Users Only! 