We begin with a definition.
Let A
be an n x n
matrix and let M A
Let
then
so
the minor of a (2)(4) = 8 The cofactor is
A 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
det A = S
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
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
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
a
a
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
a 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 = b and the theorem is proven. Let A
be an n x n
matrix. Then the
(adj A) 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) I
The proof follows immediately from the formula for the determinant and the previous theorem. We have
[A(adj A)] where
dij The main application of this theorem is the following corollary that easily follows from the theorem.
If A is a nonsingular matrix then
1
We found that the matrix
has adjoint
We can find that
det A = [A(adj A)] 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
A matrix is nonsingular if and only if the determinant is nonzero.
If A is nonsingular, then
1 = det(I) = det(AA 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.
TFAE - A
is nonsingular.
- A
**x**= 0 has only the trivial solution.
- A
is row equivalent to the identity.
- A
**x**=**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.
Let A be a linear system of equations with n x n matrix A. Then the solution is
det(A where
A
x - 3y + 2z = 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
A
We find z by dividing
20
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