Define the Identity matrix.

The Identity matrix is a square matrix with 1's on the main diagonal and 0's elsewhere.

The Identity matrix, denoted by I, is a square matrix with the same number of rows and columns. It has 1's on the main diagonal and 0's elsewhere. For example, the 3x3 Identity matrix is:

I = [1 0 0]
[0 1 0]
[0 0 1]

The Identity matrix is a special matrix in that it behaves like the number 1 in multiplication. That is, when a matrix A is multiplied by the Identity matrix I, the result is the matrix A itself. This is known as the identity property of matrix multiplication. Mathematically, we can write:

AI = IA = A

where A is any matrix of appropriate size.

The Identity matrix is also important in linear algebra, where it is used to find the inverse of a matrix. The inverse of a matrix A, denoted by A^-1, is a matrix such that when A is multiplied by A^-1, the result is the Identity matrix I. Mathematically, we can write:

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

However, not all matrices have an inverse, and the Identity matrix is only used to find the inverse of invertible matrices.