Inverse of a Matrix – Using Minors, Cofactors and Adjugate

# Inverse of a Matrix – Using Minors, Cofactors and Adjugate

## Inverse of Matrix

The inverse of a matrix is a matrix that “undoes” the original matrix. To find the inverse of a matrix, you must first determine its determinant. If the determinant is not zero, the inverse matrix can be found using the inverse matrix formula.

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## What is the Matrix of Cofactors?

A matrix of cofactors is a square matrix that is used to calculate the determinant of a larger matrix. The matrix of cofactors is created by taking the cofactor of each element in a larger matrix. The cofactor of an element is the determinant of the matrix that is created by taking the element out of the larger matrix.

## Matrix of Cofactors Formula

The matrix of cofactors formula calculates the determinant of a square matrix using the cofactors of the matrix. The cofactors of a matrix are the elements of the matrix that are located on the main diagonal. The matrix of cofactors formula is:

D = det(A) = (a ij )

Where:

D is the determinant of the matrix.

A is the square matrix.

a ij is the cofactor of the element in the ith row and the jth column of the matrix.

## Matrix of Cofactors Example

The following is a matrix of cofactors for a 4×4 matrix:

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

The adjugate matrix is a matrix that is created by adding the transpose of a matrix to the matrix itself. This matrix is used to solve systems of linear equations.

## Minor of Matrices

A minor of a matrix is a matrix that is obtained by deleting one or more rows and/or columns from the original matrix.

## Minors of Matrix Example

Let’s take a look at the minors of the matrix A =

A =

[1 2 3] [4 5 6]

The minors of A are the matrices formed by deleting any row or column from A.

For example, the minor of A with respect to the first row is

A1 =

[1 2 3] [4 5 6]

The minor of A with respect to the first column is

A1 =

[1 2 3] [4 5 6]

The minor of A with respect to the second row is

A2 =

[2 3] [4 5 6]

The minor of A with respect to the second column is

A2 =

[2 3] [4 5 6]

## Finding Inverse of Matrix Using Minors, Cofactors, and Adjugate?

The inverse of a matrix can be found using minors, cofactors, and the adjugate. First, the minors of the matrix are found. Next, the cofactors of the matrix are found. Finally, the adjugate is used to find the inverse of the matrix.

## Inverse of Matrix Using Minors, Cofactors, and Adjugate Example

Given the matrix:

A =

We can use minors, cofactors, and the adjugate to find the inverse matrix.

First, we find the minors of A.

The minors of A are:

A1 =

A2 =

A3 =

A4 =

A5 =

A6 =

A7 =

A8 =

A9 =

A10 =

Next, we find the cofactors of A.

The cofactors of A are:

A1 =

A2 =

A3 =

A4 =

A5 =

A6 =

A7 =

A8 =

A9 =

A10 =

Finally, we find the adjugate of A.

A1 =

A2 =

A3 =

A4 =

A5 =

A6 =

A7 =

A8 =

A9 =

A10 =

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