# NCERT Class 12-Mathematics: Exemplar Chapter –4 Determinants Part 2

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4.1.7 Adjoint and inverse of a matrix

(i) The adjoint of a square matrix is defined as the transpose of the matrix

, where is the co-factor of the element . It is denoted by A.

If is co-factor of

(ii) I, where A is square matrix of order .

(iii) A square matrix A is said to be singular or non-singular according as or , respectively.

(iv) If A is a square matrix of order n, then .

(v) If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.

(vi) The determinant of the product of matrices is equal to product of their respective determinants, that is, .

(vii) If , where A and B are square matrices, then B is called inverse of A and is written as . Also .

(viii) A square matrix A is invertible if and only if A is non-singular matrix.

(ix) If A is an invertible matrix, then

4.1.8 System of linear equations

(i) Consider the equations:

In matrix form, these equations can be written as , where

(ii) Unique solution of equation is given by , where .

(iii) A system of equations is consistent or inconsistent according as its solution exists or not.

(iv) For a square matrix A in matrix equation

(a) If , then there exists unique solution.

(b) If and , then there exists no solution.

(c) If and , then system may or may not be consistent.

## 4.2 Solved Examples

Question 1:

If , Then find

We have . This gives

.

Question 2:

If , then prove that

We have

Interchanging rows and columns, we get

Interchanging and

Question 3:

Without expanding, show that

Applying , we have

Question 4:

Show that