Inverse of matrix

Jun 12, 2020, 16:45 IST

Inverse of matrix

Definition :-Assuming that we have a square matrix a, which is non-singular (i.e. Det (a) does not equal zero), then there exists an n × n matrix. which is called the inverse of a such that:
where i is the identity matrix.

The inverse of a 2×2 matrix
take for example an arbitrary 2×2 matrix a whose determinant (ad − bc) is not equal to zero.where a, b, c and d are numbers.
The inverse is:the inverse of a general n × n matrix a can be found by using the following equation.where the adj (a) denotes the adjoint of a matrix. It can be calculated by the following method:

given the n × n matrix a, define b = bij to be the matrix whose coefficients are found by taking the determinant of the (n-1) × (n-1) matrix obtained by deleting the ith row and jth column of a.

The terms of b (i.e. B = bij) are known as the cofactors of a.
Define the matrix c, where

The transpose of c (i.e. Ct) is called the adjoint of matrix a.

Example 1 :-Find the inverse of

Solution :-Hence exists.
Cofactors of A are:

Example 2 :-Find the inverse of the matrix

Solution :-Here,Expanding using 1st row, we get

= 1(6 –1) –2(4 –3) + 3(2 – 9)
= 5 – 2 × 1 + 3 × (–7)
= 5 – 2 – 21 = – 180

Therefore,exists.

Cofactors of A are:

Do solve NCERT text book with the help of Entrancei NCERT solutions for class 12 Maths.