Inverse matrix is a matrix also called inverse transformation matrix, which undoes the transformation done by original matrix. Transformation means matrix multiplication.

Inverse of a matrix A is denoted by A^{-1}.

Finding the inverse of a matrix is very simple. Just multiply the adjoing of a matrix with 1 divided by determinant of matrix.

Formula:

```
\(D = \frac{1}{\text{det}(M)}
\)
```

```
\(M^{-1} = D \cdot \text{adj}(M)
\)
```

where,

- M^-1 is the inverse of matrix M

Its the same formula for 2x2 matrix as well as 3x3 matrix. The only difference is in the calculation of adjoint and determinant, which becomes more time and resource consuming as the order of matrix increases.

Therefore, we use matrix calculators to perform these matrix operations

Imagine a matrix M which transforms a vector K into a vector L.

```
\(L = M \cdot K
\)
```

Now the inverse of matrix M, denoted by M^{-1}, will transform the vector L back to K.

```
\(K = M^{-1} \cdot L
\)
```

The matrix M^{-1} is undoes the transformation done by matrix M. Its is like a reverse transformation.

Identity Matrix is a matrix that does nothing. If a vector K goes through a transformation with identity matrix, it remains same.

```
\(K = I \cdot K
\)
```

Identity Matrix is born when a matrix is multiplied with its inverse.

If M and M^{-1} are multiplied together, the result is an identity matrix. Identity matrix is a matrix that does nothing.

```
\(
M \cdot M^{-1} = I
\)
```

where,

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

```
\(
M = \text{square matrix of order 2}
\)
```

There are set of rules that tells whether a matrix is invertible or not. These rules are called invertible matrix theorem.

There are different variation of invertible matrix theorem. But the idea behind them is always the same.

The basic rules are as follows:

- Matrix should have uniform dimension (equal number of rows and columns)
- Matrix should have non-zero determinant
- Matrix should be full rank

Other rules are just extension of these basic rules.

Not all matrices have inverse. There are some matrices that do not have inverse.

**Singular Matrix**

Singular matrices are square matrix that do not have inverse. Singular matrices have determinant of zero. Singular matrices are also called degenerate matrices.

**Non-Square Matrix**

Non-square matrices do not have inverse. Its because they don't have equal number of rows and columns.

The properties of matrix inverse are:

- The inverse of inverse of matrix is always the matrix itself.
- A invertible matrix have only one inverse.
- The inverse of (transpose of matrix) is the transpose of matrix's inverse
- Identity Matrix is the inverse of itself.

Inverse of a matrix is very useful in different fields of mathematics and science. Some of the uses are:

- Computer Graphics and Animation
- Stastics and Data Analysis
- Engineering and Physics
- Cryptography

The inverse of a identity matrix is itself.

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