Inverse of Non Square Matrix

Introduction to inverse of non square matrix:

Matrix has a list of data. In math matrix is a rectangular arrangement of the elements. The elements are shown in the rows and the columns. In math array elements are put in the parenthesis or square brackets. In math matrix is represented by capital letters for example A, B, C…… Square matrix has equal number of rows and equal number of column. If matrix has not equal number of rows and columns called as non square matrix.

Inverse of Non Square Matrix:

Square matrix:

Square matrix has equal number of rows and equal number of columns.

Example:

`[[a,b],[c,d]]` 

The above matrix has equal number of rows and equal number of columns. So it is called as square matrix. The order of square matrix is represented by n `xx` n or m `xx` m. The order of above matrix is 2 `xx` 2.

Non-square matrix:

Non square matrix has not equal number of rows and columns.

Example:

`[[a,b],[c,d],[e,f]]` 

The above matrix has 3 rows and 2 columns. The number of rows and number of columns of given matrix is not equal so it is a non-square matrix.

Inverse of non square matrix:

Here we see additive inverse of non-square matrix. The additive inverse of matrix X is –X.

In additive inverse put – sign to all the positive numbers in the given matrix and put + sigh to all the negative number in the given matrix. The addition of normal and inverse matrix is zero matrixes.

Additive rules for inverse of non-square matrix.

X + (-X) = (-X) + X = 0

Example:

Matrix A:

A= `[[2,1],[6,4],[9,7]]`

Inverse of matrix A:

-A = `[[-2,-1],[-6,-4],[-9,-7]]`

Example Sums for Inverse of Non Square Matrix:

Example 1:

A= `[[-5,2],[3,-8],[-1,-4]]`

Find inverse of matrix A

Solution:

Given A=  `[[-5,2],[3,-8],[-1,-4]]`

In additive inverse put – sign to all the positive numbers in the given matrix and put + sigh to all the negative number in the given matrix. The addition of normal and inverse of a matrix is zero matrixes.

-  A= -  `[[-5,2],[3,-8],[-1,-4]]`

- A =  `[[5,-2],[-3,8],[1,4]]`

Example 2:

Prove

X + (-X) = (-X) + X = 0

Solution:

Let take X = `[[1,2],[3,4],[5,6]]`

- X = -  `[[1,2],[3,4],[5,6]]`

- X= `[[-1,-2],[-3,-4],[-5,-6]]`

X + (-X) =  `[[1,2],[3,4],[5,6]]` + `[[-1,-2],[-3,-4],[-5,-6]]`

X + (-X) = 0

(-X) + X is similar to X + (-X)

Therefore X + (-X) = (-X) + X = 0 is proved