Cofactors and Determinants

We begin with a 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







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.



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. 



Use cofactors to find det A for



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.


Find the determinant of



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.



Let A be an n x n matrix.  Then


        ai1Ak1 +  ai2Ak2 + ... +  ainAkn  =  0     for i k

        a1jA1k +  a2jA2k + ... +  anjAnk  =  0     for j k



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. 


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.



Find adj A for



We have


So that


Now for the main theorem


If A is an n x n matrix then

        A(adj A)  =  (adj A)A  =  (det A) In 



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.



If A is a nonsingular matrix then

        A-1  =                  adj A
                     det A


We found that the matrix


has adjoint


We can find that

        det A  =  [A(adj A)]11  =  27




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



A matrix is nonsingular if and only if the determinant is nonzero.



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.




  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


        Ax  =  b

be a linear system of equations with n x n matrix A.  Then the solution is

        xi  =                     

where Ai is the matrix obtained from A by replacing the ith column of A by b.



Use Cramer's rule to find z if

        x - 3y + 2z  =  3
        2x + y + z  =  1
        x + y + z  =  3


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

        z  =          =  4


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