Cofactors

Cofactors and Determinants

We begin with a definition.

Definition

Let A be an n x n matrix and let M_ij be the (n - 1) x (n - 1) matrix obtained by deleting the i^th row and j^th column.  Then det M_ij is called the minor of a_ij.  The cofactor A_ij of a_ij is defined by

        A_ij  =  (-1)^i+j det M_ij

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  =  S_j (a_ijA_ij)  =  a_i1A_i1 +  a_i2A_i2 + ... +  a_inA_in

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 A^T, 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

        a_i1A_k1 +  a_i2A_k2 + ... +  a_inA_kn  =  0     for i k

        a_1jA_1k +  a_2jA_2k + ... +  a_njA_nk  =  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 + ... +  a_1nA_2n  =  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 + ... +  b_2nB_2n  =   a₁₁A₂₁ +  a₁₂A₂₂ + ... +  a_1nA_2n 

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  =  A_ji

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) I_n 

Proof

The proof follows immediately from the formula for the determinant and the previous theorem.  We have

        [A(adj A)]_ij  =  S_k a_ik(adj A)_kj  =  S_k a_ikA_jk  =  (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

1. A is nonsingular.
   
2. Ax  =  0  has only the trivial solution.
   
3. A is row equivalent to the identity.
   
4. Ax  =  b has a unique solution for all b.
   
5. 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(A_i)
        x_i  =                     
                      det(A)

where A_i is the matrix obtained from A by replacing the i^th 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 Matrices and Applications Home Page

Back to the Linear Algebra Home Page

Back to the Math Department Home Page

e-mail Questions and Suggestions