Inverse Matrix

Matrices are an important topic in terms of class 11 mathematics. In that, most weightage is given to inverse matrix problems. Let us first define the inverse of a matrix.

A matrix is a definite collection of elements arranged in the form of rows and columns. The matrix is written in the order of the number of rows by the number of columns. For example, 2 × 2, 2 × 3, 3 × 2, 3 × 3, 4 × 4, etc. The matrix inverse can be only applicable for square matrices, in which the number of rows and the number of columns are equal. For example, 2 × 2, 3 × 3, 4 x 4, etc. 

Matrix Inverse

If there is a non-singular square matrix A, then there is a possibility for the A⁻¹ n x n matrix, which is called the inverse matrix of A.

AA⁻¹ = A⁻¹A = I, where I is called the Identity matrix.

Rank of the Matrix 

The rank of the matrix is the extreme number of linearly self-determining column vectors within the matrix.

How to find the Inverse of a Matrix/ how to Determine the Inverse of a Matrix?

The inverse matrix can be found only with the square matrix. The square matrix has to be non-singular, i.e, its determinant has to be non-zero. A common question arises, how to find the inverse of a square matrix? By inverse matrix definition in math, we can only find inverses in square matrices.

Given a square matrix A

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Its determinant value is given by

(a∗d)−(c∗d)

Inverse Matrix Formula

The inverse matrix formula is given as,

A-1 = \[ \frac{adj(A)}{|A|} \] ; |A| ≠ 0

where,

A is a square matrix.

Note: For an inverse matrix to exist:

The matrix given must be a square matrix.

In the given matrix the determinants should not be zero.

For a Matrix A= \[\begin{bmatrix}a _{11} & a _{12} & a _{13} \\a _{21} & a _{22} & a _{23} \\ a_{31} & a _{32} & a _{33} \end{bmatrix}\]

The minor of the element a11 is \[\begin{bmatrix}a _{22} & a _{23}  \\  a _{32} & a _{33} \end{bmatrix}\]

Methods to Find Inverse of Matrix

The inverse of the matrix is generally found using two elementary operations methods. The elementary operations on a matrix are performed either through row or column transformations. 

Elementary Row Operations

To calculate the inverse of a matrix through elementary row operations, take three square matrices X, A, and B respectively. 

The matrix equation is A X = B. ⇒ X = A-1 B.

Express the given matrix A as, A = I A. 

Perform the elementary row operations on both sides and obtain an identity matrix of the same.

The final matrix in R.H.S. with "A" after transformations are the inverse of the given matrix.

Elementary Column Operations

To calculate the inverse of a matrix through elementary column operations, take three square matrices X, A, and B respectively.

The matrix equation is A X = B. ⇒ X = A-1 B. 

Perform the elementary row operations, 

Express the given matrix A as, A = I A. 

Perform the elementary column operations on both sides and obtain an identity matrix of the same.

The final matrix in R.H.S. with "A" after transformations are the inverse of the given matrix.

How are we Going to Measure the Inverse?

We can calculate the inverse of the matrix in the following steps-

1st Step - First we need to check if the determinant is not equal to 0. Then Calculate a minor matrix.

2nd Step - Then convert it to a cofactor matrix.

3rd Step - After that we need to find the Adjugate or Adjoint of matrix by taking the transpose of cofactor matrix

4th Step  - Finally, multiply with 1 / Determinant.