How to calculate the inverse of a matrix?

To calculate the inverse of a matrix, use the formula A^-1 = 1/det(A) * adj(A), where det(A) is the determinant of A and adj(A) is the adjugate of A.

To find the determinant of a matrix, use the formula det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31), where aij represents the element in the ith row and jth column of A.

To find the adjugate of a matrix, first find the matrix of cofactors by taking the determinant of each minor matrix (the matrix formed by deleting the ith row and jth column of A) and multiplying it by (-1)^(i+j). Then transpose the resulting matrix to get the adjugate.

Example: Find the inverse of the matrix A = [2 1; 4 3]

det(A) = 2(3) - 1(4) = 2
adj(A) = [3 -4; -1 2]
A^-1 = 1/2 * [3 -4; -1 2] = [3/2 -2; -1/2 1]