# Properties of Matrices Inverse

If A is a non-singular square matrix, there is an existence of n x n matrix A^{-1}, which is called the inverse of a matrix A such that it satisfies the property:

AA^{-1} = A^{-1}A = I, where I is the Identity matrix

The identity matrix for the 2 x 2 matrix is given by

\(I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\)

It is noted that in order to find the matrix inverse, the square matrix should be non-singular whose determinant value does not equal to zero.

Let us take the square matrix A

\(A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\)

Where a, b, c, and d represents the number.

The determinant of the matrix A is written as ad-bc, where the value is not equal to zero. In this article, let us discuss the important properties of matrices inverse with example.

**Also, read:**

## Matrix Inverse Properties

The list of properties of matrices inverse is given below. Go through it and simplify the complex problems.

If A and B are the non-singular matrices, then the inverse matrix should have the following properties

- (A
^{-1})^{-1}=A - (AB)
^{-1}=A^{-1}B^{-1} - (ABC)
^{-1}=C^{-1}B^{-1}A^{-1} - (A
_{1 }A_{2}….A_{n})^{-1}=A_{n}^{-1}A_{n-1}^{-1}……A_{2}^{-1}A_{1}^{-1} - (A
^{T})^{-1}=(A^{-1})^{T} - (kA)
^{-1}= (1/k)A^{-1} - AB = I
_{n}, where A and B are inverse of each other. - If A is a square matrix where n>0, then (A
^{-1})^{n }=A^{-n}

Where A^{-n} = (A^{n})^{-1}

## Solved Example

The example of finding the inverse of the matrix is given in detail. Go through it and learn the problems using the properties of matrices inverse.

**Question:**

Find the inverse A^{-1} of the matrix \(A=\begin{bmatrix} 2 & 1 & 1\\ 3 & 2 & 1\\ 2 & 1 & 2 \end{bmatrix}\)

**Solution:**

Given: \(A=\begin{bmatrix} 2 & 1 & 1\\ 3 & 2 & 1\\ 2 & 1 & 2 \end{bmatrix}\)

We know that

A^{-1} = adj (A) / det(A)

A^{-1} = adj (A) / |A|

|A| = 2(4-1) -1(6-2)+1(3-4)

|A| = 2(3) -1(4)+1(-1)

|A| = 6-4-1

|A| = 1

Now, find Adj(A):

\(A_{c}=\begin{bmatrix} (4-1) & -(6-2) & (3-4) \\ -(2-1) &(4-2) &-(2-2) \\ (1-2) &-(2-3) & (4-3) \end{bmatrix}\) \(A_{c}=\begin{bmatrix} 3 & -4 & -1 \\ -1 &2 &0 \\ -1 &1 & 1 \end{bmatrix}\)Now, take the transpose of the cofactor matrix

\(A_{c}^{T}=\begin{bmatrix} 3 & -1 & -1 \\ -4 &2 &1 \\ -1 &0 & 1 \end{bmatrix}\)Therefore,

A^{-1} = adj (A) / |A|

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