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