## Inverse of A Square Matrix

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# Inverse of A Square Matrix

Definition. Let A be a square matrix of order n. Then a square matrix B of order n, if it exists, is called an inverse of A if AB = BA = In.

A matrix A having an inverse is called an invertible matrix. It may easily be seen that if a matrix A is invertible, its inverse is unique. The inverse of a invertible matrix A is denoted by A

Does every square matrix possess an inverse To answer this let us consider the matrix

If B is any square matrix of order 2, we find that

We thus see that there cannot be any matrix

However, if

A

|A|

For, we know that

Thus A is invertible and A

|A|

Singular and Non-singular Matrices. A square matrix A is said to be singular if |A| = 0, and it is called nonsingular if |A| ≠ 0. For example, if

then |A| = 3 (-3+2) - 1 (2+1) +2 (4+3) = 8.

Since |A| ≠ 0, A is non-singular. Further, if

then |B| = 2 (-8+8) + 0 + 0 =0

Since |B| = 0, B is singular

1. A square matrix is invertible if and only if it is non-singular.

2. The inverse of the inverse is the original matrix itself, i.e. (A

3. The inverse of the transpose of a matrix is the transpose of its inverse, i.e. (A')

4. (Reversal law) if A and B are two invertible matrices of the some order, then AB is also invertible and moreover.

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A matrix A having an inverse is called an invertible matrix. It may easily be seen that if a matrix A is invertible, its inverse is unique. The inverse of a invertible matrix A is denoted by A

^{-1}.Does every square matrix possess an inverse To answer this let us consider the matrix

If B is any square matrix of order 2, we find that

**AB = BA = 0.**We thus see that there cannot be any matrix

**B**for which**AB**and**BA**both are equal to**I**. Therefore A is not invertible. Hence, we con-clued that a square matrix may fail to have an inverse._{2}However, if

**A**is square matrix such that |**A**| ≠ 0, then A is invertible andA

^{-1}=__1__adj A.|A|

For, we know that

**A**(adj A) = (adj A)**A**= |**A**| InThus A is invertible and A

^{-1 }=__1__adj A|A|

Singular and Non-singular Matrices. A square matrix A is said to be singular if |A| = 0, and it is called nonsingular if |A| ≠ 0. For example, if

then |A| = 3 (-3+2) - 1 (2+1) +2 (4+3) = 8.

Since |A| ≠ 0, A is non-singular. Further, if

then |B| = 2 (-8+8) + 0 + 0 =0

Since |B| = 0, B is singular

**Properties of the Inverse of a Matrix**1. A square matrix is invertible if and only if it is non-singular.

2. The inverse of the inverse is the original matrix itself, i.e. (A

^{-1})^{-1}=A3. The inverse of the transpose of a matrix is the transpose of its inverse, i.e. (A')

^{-1}= (A^{-1})'.4. (Reversal law) if A and B are two invertible matrices of the some order, then AB is also invertible and moreover.

**(AB)**^{-1}= B^{-1}A^{-1}.For more help in

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