Cofactors Cofactors and Determinants We begin with a definition. Definition Let A be an n x n matrix and let Mij be the (n - 1) x (n - 1) matrix obtained by deleting the ith row and jth column.  Then det Mij is called the minor of aij.  The cofactor Aij of aij is defined by         Aij  =  (-1)i+j det Mij   Example Let then so the minor of a32 is the determinant of this 2 x 2 matrix.  Since the matrix is triangular, the determinant is the product of the diagonals or         (2)(4)  =  8 The cofactor is         A23  =  (-1)2+3(8)  =  -8 One of the main applications of cofactors is finding the determinant.  The following theorem, which we will not prove, shows us how to use cofactors to find a determinant.   Theorem Let A be an n x n matrix and 1 < i < n.  Then         det A  =  Sj (aijAij)  =  ai1Ai1 +  ai2Ai2 + ... +  ainAin   This theorem has little meaning without an example.    Example Use cofactors to find det A for Solution We can use any row that we want.  Let's pick the second row.  We have =  0 + (3)(0 - 5) + (8 - 0)  =  -7   Remark:  Since det A  =  det AT, we can expand about a column if we desire. Example Find the determinant of Solution We can choose any row or column to expand.  The third column has only one nonzero entry, so we select this column.  We have Now lets expand about the third row.  We get Cofactors and Inverses Just as cofactors can be used to find the determinant of a matrix, they can be used to find the inverse of a matrix.  We begin with a theorem that will be useful for proving the inverse formula.   Theorem Let A be an n x n matrix.  Then                 ai1Ak1 +  ai2Ak2 + ... +  ainAkn  =  0     for i k         a1jA1k +  a2jA2k + ... +  anjAnk  =  0     for j k   Proof We will prove the first statement for i = 1 and k = 2.  The general case and the second statement can be proven in a similar way.  We want to show that         a11A21 +  a12A22 + ... +  a1nA2n  =  0  Consider the matrix B that is the same as A except that the second row of B is the same as the first row.  Since two rows of B are repeated, the determinant of B is zero.  Now find det B by expanding about the second row.  You will notice that this expansion is         0  =  b21B21 +  b22B22 + ... +  b2nB2n  =   a11A21 +  a12A22 + ... +  a1nA2n  and the theorem is proven.  Definition Let A be an n x n matrix. Then the adjoint of A (adj A) is the matrix such that         (adj A)ij  =  Aji Notice the switch of subscripts.  This means that the adjoint is the transpose of the matrix that consists of cofactors.   Example Find adj A for Solution We have So that Now for the main theorem Theorem If A is an n x n matrix then         A(adj A)  =  (adj A)A  =  (det A) In    Proof The proof follows immediately from the formula for the determinant and the previous theorem.  We have         [A(adj A)]ij  =  Sk aik(adj A)kj  =  Sk aikAjk  =  (det A)dij    where dij  is the Kronecker delta function evaluating to 1 for i = j and 0 otherwise.  Hence the theorem is proven. The main application of this theorem is the following corollary that easily follows from the theorem.   Corollary If A is a nonsingular matrix then                         1         A-1  =                  adj A                      det A Example We found that the matrix has adjoint We can find that         det A  =  [A(adj A)]11  =  27 Hence This gives us a way to find inverses and a way to determine if a matrix is nonsingular.  For a 2 x 2 matrix the adjoint of A is easy to find.  We have Using the inverse formula, we get Theorem A matrix is nonsingular if and only if the determinant is nonzero.   Proof If A is nonsingular, then          1  =  det(I)  =  det(AA-1)  =  (det A) (det A-1) so the determinant of A is nonzero. If the determinant is nonzero, then the corollary shows us how to find the inverse so the matrix is nonsingular. This gives us an addition to our list of nonsingular equivalences.   Theorem TFAE A is nonsingular. Ax  =  0  has only the trivial solution. A is row equivalent to the identity. Ax  =  b has a unique solution for all b. The determinant of A is nonzero. We end this discussion with the statement of Cramer's Rule, a formula that gives us the solution of systems of equations. Cramer's Rule Let         Ax  =  b be a linear system of equations with n x n matrix A.  Then the solution is                       det(Ai)         xi  =                                            det(A) where Ai is the matrix obtained from A by replacing the ith column of A by b.   Example         Use Cramer's rule to find z if         x - 3y + 2z  =  3         2x + y + z  =  1         x + y + z  =  3 Solution We write this in matrix form We have         det A  =  1(1 - 1) - (-3)(2 - 1) + 2(2 - 1)  =  5 since we want to find z, we need det A3. We find z by dividing                  20         z  =          =  4                   5         Back to the Linear Algebra Home Page