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-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 I2. Therefore A is not invertible. Hence, we con-clued that a square matrix may fail to have an inverse.
However, if A is square matrix such that |A| ≠ 0, then A is invertible and
A-1 = 1 adj A.
|A|
For, we know that
A (adj A) = (adj A) A = |A| In
Thus 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 =A
3. 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 .
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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 I2. Therefore A is not invertible. Hence, we con-clued that a square matrix may fail to have an inverse.
However, if A is square matrix such that |A| ≠ 0, then A is invertible and
A-1 = 1 adj A.
|A|
For, we know that
A (adj A) = (adj A) A = |A| In
Thus 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 =A
3. 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 Inverse of A Square Matrix click the button below to submit your homework assignment