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When we first introduced the determinant we motivated its definition for a \(2\times 2\) matrix by the fact that the value
of the determinant is zero if and only if the matrix is singular. We will soon be able to generalize this result
to larger matrices, and will eventually establish a formula for the inverse of a nonsingular matrix in terms of
determinants.
Recall that we can find the inverse of a matrix or establish that the inverse does not exist by using elementary row operations
to carry the given matrix to its reduced row-echelon form. In order to start relating determinants to inverses we need to find
out what elementary row operations do to the determinant of a matrix.
The Effects of Elementary Row Operations on the Determinant
Recall that there are three elementary row operations:
1.
Switching the order of two rows
2.
Multiplying a row by a non-zero constant
3.
Adding a multiple of one row to another
Elementary row operations are used to carry a matrix to its reduced row-echelon form. In Practice Problem ?? we established
that elementary row operations are reversible. In other words, if we know what elementary row operations carried \(A\) to \(\mbox {rref}(A)\), we can
undo each operation with another elementary row operation to carry \(\mbox {rref}(A)\) back to \(A\). This will prove useful for computing the
determinant. Computing the determinant of \(\mbox {rref}(A)\) is easy. (Why?) If we know what elementary row operations carry \(\mbox {rref}(A)\) back
to \(A\), and what effect each of these operations has on the determinant of \(\mbox {rref}(A)\), we could find the determinant of
\(A\).
This result is particularly surprising. Try a few more variations of this example to convince yourself that adding a multiple of
one row to another row does not appear to affect the determinant.
The following theorem generalizes our observations.
Let \(A=\begin{bmatrix}a_{ij}\end{bmatrix}\) be an \(n\times n\) matrix.
1.
If \(B\) is obtained from \(A\) by interchanging two different rows, then
\[\det {B}=-\det {A}\]
2.
If \(B\) is obtained from \(A\) by multiplying one of the rows of \(A\) by a non-zero constant \(k\). Then
\[\det {B}=k\det {A}\]
3.
If \(B\) is obtained from \(A\) by adding a multiple of one row of \(A\) to another row, then
The following lemma is a useful consequence of parts 1 and 2 of Theorem 1.
Let \(A\) be an \(n\times n\) matrix.
1.
If \(A\) has a row of zeros, then \(\det {A}=0\).
2.
If two rows of \(A\) are the same, then \(\det {A}=0\).
3.
If one row of \(A\) is a scalar multiple of another row, then \(\det {A}=0\).
We will prove Part 2. Parts 1 and 3 are left as exercises.
Proof of Part 2 Suppose rows \(p\) and \(q\) of \(A\) are the same. Let \(B\) be a matrix obtained from \(A\) by switching \(p\) and \(q\). By Theorem 11 we
know that \(\det {B}=-\det {A}\). But \(p\) and \(q\) are the same, so \(A=B\). But then \(\det {A}=-\det {A}\). We conclude that \(\det {A}=0\).
Because \(\det {A}=\det {A^T}\), we have the following counterpart of Theorem 1 for columns.
Elementary Column Operations and the Determinant Let \(A\) be an \(n\times n\) matrix.
1.
If \(B\) is obtained from \(A\) by interchanging two different columns, then
\[\det {B}=-\det {A}\]
2.
If \(B\) is obtained from \(A\) by multiplying one of the columns of \(A\) by a non-zero constant \(k\). Then
\[\det {B}=k\det {A}\]
3.
If \(B\) is obtained from \(A\) by adding a multiple of one column of \(A\) to another column, then
\[\det {B}=\det {A}\]
Computing the Determinant Using Elementary Row Operations
What we discovered about the effects of elementary row operations on the determinant will allow us to compute determinants
without using the cumbersome process of cofactor expansion.
Suppose that a \(6\times 6\) matrix \(A\) is carried to the identity matrix by a sequence of elementary row operations listed below. Find
\(\det {A}\).
