Define the inverse of a matrix.

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.

To find the inverse of a matrix A, we must first check if it is invertible. A matrix is invertible if its determinant is not equal to zero. If det(A) = 0, then A does not have an inverse.

If A is invertible, we can find its inverse by using the formula A^-1 = (1/det(A)) adj(A), where adj(A) is the adjugate of A. The adjugate of A is the transpose of the matrix of cofactors of A.

For example, let A = [2 1; 4 3]. To check if A is invertible, we calculate det(A) = (2*3) - (1*4) = 2. Since det(A) is not equal to zero, A is invertible.

To find A^-1, we first calculate the adjugate of A: adj(A) = [3 -1; -4 2]. Then, we use the formula A^-1 = (1/det(A)) adj(A) = (1/2) [3 -1; -4 2] = [3/2 -1/2; -2 1].

We can check that A^-1 is indeed the inverse of A by multiplying A and A^-1:

A * A^-1 = [2 1; 4 3] * [3/2 -1/2; -2 1] = [1 0; 0 1] = I, the identity matrix.