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 a_{32} 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 A_{23} = (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_{ij}A_{ij}) = a_{i1}A_{i1} + a_{i2}A_{i2} + ... + a_{in}A_{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_{i1}A_{k1} + a_{i2}A_{k2} + ... + a_{in}A_{kn} = 0 for i k a_{1j}A_{1k} + a_{2j}A_{2k} + ... + a_{nj}A_{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 a_{11}A_{21} + a_{12}A_{22} + ... + a_{1n}A_{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 = b_{21}B_{21} + b_{22}B_{22} + ... + b_{2n}B_{2n} = a_{11}A_{21} + a_{12}A_{22} + ... + a_{1n}A_{2n} 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} = 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}
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_{ik}A_{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 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
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}) where A_{i} is the matrix obtained from A by replacing the i^{th} column of A by b.
Example
x  3y + 2z = 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 A_{3}.
We find z by dividing
20
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