Calculation of Inverse Matrix:



Calculation of Inverse Matrix:

1. Using Gauss-Jordan reduction:

The procedure for computing the inverse of a [pic] matrix A:

1. Form the [pic] augmented matrix

[pic]

and transform the augmented matrix to the matrix

[pic]

in reduced row echelon form via elementary row operations.

2. If

(a) [pic], then [pic].

(b) [pic], then [pic] is singular and [pic] does not exist.

Example:

To find the inverse of [pic], we can employ the procedure introduced above.

1.

[pic].

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

2. The inverse of A is

[pic].

Example:

Find the inverse of [pic]if it exists.

[solution:]

1. Form the augmented matrix

[pic].

And the transformed matrix in reduced row echelon form is

[pic]

2. The inverse of A is

[pic].

Example:

Find the inverse of [pic]if it exists.

[solution:]

1. Form the augmented matrix

[pic].

And the transformed matrix in reduced row echelon form is

[pic]

2. A is singular!!

2. Using the adjoint [pic] of a matrix:

As [pic], then

[pic].

Note:

[pic] is always true.

Note:

As [pic] [pic] A is nonsingular.

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