Now, since Laplace expansion along the first row gives Įxample 2: Determine the inverse of the following matrix by first computing its adjoint:įirst, evaluate the cofactor of each entry in A: A square matrix A is invertible if and only if its determinant is not zero, and its inverse is obtained by multiplying the adjoint of A by (det A) −1. This result gives the following equation for the inverse of A:īy generalizing these calculations to an arbitrary n by n matrix, the following theorem can be proved:
Now, since a Laplace expansion by the first column of A gives Why form the adjoint matrix? First, verify the following calculation where the matrix A above is multiplied by its adjoint:
The first step is to evaluate the cofactor of every entry: The transpose of the matrix whose ( i, j) entry is the a ijcofactor is called the classical adjoint of A:Įxample 1: Find the adjoint of the matrix
