How do you find the inverse of an identity matrix?

The inverse of an identity matrix is the identity matrix itself.

To understand this, let's revisit the concept of an inverse matrix. The inverse of a matrix, when multiplied by the original matrix, results in the identity matrix. According to the provided reference, we determine the inverse of an identity matrix by finding a matrix that, when multiplied by the identity matrix, equals the identity matrix. Logically, this is the identity matrix itself.

Understanding Identity Matrices

An identity matrix is a special square matrix where all the elements on the main diagonal are 1, and all other elements are 0. For example, a 3x3 identity matrix looks like this:

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

Key properties of identity matrices are:

Any matrix multiplied by the identity matrix of appropriate dimensions results in the same matrix. This is similar to how multiplying a number by 1 results in the same number.
Identity matrices are always square (number of rows equals the number of columns).

Finding the Inverse

Let's denote the identity matrix as *I*. The inverse of matrix *A* is written as *A^-1*. The defining property of inverse matrices is:

A A^-1 = A^-1 A = I

Where *I* is the identity matrix.

In the case of the identity matrix itself, we're looking for a matrix (*I^-1*) that satisfies:

*I* *I^-1* = *I^-1* *I* = *I*

Because multiplying any matrix by the identity matrix results in the same matrix, it follows that:

*I* *I* = *I*

This means that the inverse of the identity matrix is the identity matrix itself.

Examples

Let's look at some examples:

2x2 Identity Matrix: If I₂ =
```
[ 1 0 ]
[ 0 1 ]
```
Then, I₂^-1 =
```
[ 1 0 ]
[ 0 1 ]
```
3x3 Identity Matrix: If I₃ =
```
[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]
```
Then, I₃^-1=
```
[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]
```
In general, no matter the size of the identity matrix, its inverse will always be the original identity matrix.

Conclusion

The inverse of an identity matrix is simply the same identity matrix. This is because multiplying an identity matrix by itself yields the identity matrix, satisfying the definition of an inverse.