Recall that if B is n x n and k is a scalar, then det( kB) = k ndet B. Which may be verified by checking that AA −1 = A −1 A = I.Įxample 3: If A is an invertible n by n matrix, compute the determinant of Adj A in terms of det A.īecause A is invertible, the equation A −1 = Adj A/det A implies 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
