Now, we define the inverse of a square matrix.

**Definition of inverse matrix of a square matrix**

Now, we define the
inverse of a square matrix.

Let *A *be a
square matrix of order *n*. If there exists a square matrix *B *of
order *n *such that *AB *= *BA *= *I _{n} *, then the matrix

If a square matrix has an inverse, then it is unique.

**Proof**

Let *A *be a square matrix order *n *such that an
inverse of *A *exists. If possible, let there be two inverses *B *and
*C *of *A*. Then, by definition, we have *AB *= *BA *= *I _{n} *and

Using these equations,
we get

*C *= *CI _{n} *=

Hence the uniqueness
follows.

**Notation **The inverse of a matrix *A *is denoted by *A*^{âˆ’}^{1}.

**Note**

*AA*^{âˆ’}^{1} = *A*^{âˆ’}^{1} *A *= *I _{n} *.

**Theorem 1.3**

Let *A *be square matrix of order *n*. Then, *A*^{âˆ’}^{1} exists if and only if
*A *is non-singular.

**Proof**

Suppose that *A*^{âˆ’}^{1} exists. Then *AA*^{âˆ’}^{1} = *A*^{âˆ’}^{1} *A *= *I _{n} *.

By the product rule for determinants, we get

det( *AA*^{âˆ’}^{1} ) =
det( *A*) det( *A*^{âˆ’}^{1} ) =
det( *A*^{âˆ’}^{1} ) det( *A*) =
det(*I _{n} *) =
1. So, |

Hence *A *is
non-singular.

Conversely, suppose that *A *is non-singular.

Then |*A | â‰ * 0. By Theorem 1.1, we
get

*A*(adj *A*) = (adj *A*) *A* = |A| I_{n}.

Thus,
we are able to find a matrix B = 1/|A| adj *A*
such that *AB* = *BA = *I_{n} .

Hence,
the inverse of *A* exists and it is
given by

The determinant of a
singular matrix is 0 and so a singular matrix has no inverse.

**Example 1.2**

If A = is non-singular, find A^{âˆ’1}

**Solution**

We first find adj *A*. By
definition, we get

Find the inverse of the matrix

Solution

Tags : Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Definition of inverse matrix of a square matrix | Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix

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