Elementary Row Operations and the Determinant

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:

(a)
Switching the order of two rows
(b)
Multiplying a row by a non-zero constant
(c)
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 \(A\).

Let
\[A=\begin{bmatrix}3&-1&-1\\3&1&-2\\-1&4&2\end{bmatrix}\]
Find \(\det {A}\).
\[\det {A}=\answer {21}\]
Construct matrix \(B\) by switching the first and the third rows of \(A\).
\[B=\begin{bmatrix}-1&4&2\\3&1&-2\\3&-1&-1\end{bmatrix}\]
Find \(\det {B}\).
\[\det {B}=\answer {-21}\]
Next, try switching consecutive rows. Construct matrix \(B'\) by interchanging the second and third rows of \(A\).
\[B'=\begin{bmatrix}3&-1&-1\\-1&4&2\\3&1&-2\end{bmatrix}\]
Find \(\det {B'}\).
\[\det {B'}=\answer {-21}\]
It appears that switching any two rows of a matrix produces a determinant that is negative of the determinant of the original matrix.

Next, construct matrix \(C\) by multiplying the last row of \(A\) by \(k\).

\[C=\begin{bmatrix}3&-1&-1\\3&1&-2\\-k&4k&2k\end{bmatrix}\]
Find \(\det {C}\).
\[\det {C}=\answer {21}k\]
It turns out that multiplying the first or the second row of \(A\) by \(k\) yields exactly the same result as this.

Finally, construct matrix \(D\) by adding twice row 3 to row 1.

\[D=\begin{bmatrix}1&7&3\\3&1&-2\\-1&4&2\end{bmatrix}\]
Find \(\det {D}\).
\[\det {D}=\answer {21}\]
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.

The proof of this theorem is relegated to Tedious Proofs Concerning Determinants. For a sketch of the proof, you can watch this video:

The following lemma is a useful consequence of parts (a) and (b) of Theorem 1.

We will prove Part (b). Parts (a) and (c) are left as exercises.

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 1(a) 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.

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.

Practice Problems

Let \(A\) be an \(n\times n\) matrix. Show that
\[\det {kA}=k^n\det {A}\]
Prove that if one row of \(A\) is a scalar multiple of another row, then \(\det {A}=0\). (Lemma 2(c))
Apply (b) of Theorem 1 to a matrix that has two identical rows.
Prove that if one row of a matrix is a linear combination of two other rows of the matrix, then the determinant of the matrix is 0.
Find \(\det {A}\) using elementary row operations.
\[A=\begin{bmatrix}3&1&-3\\12&5&-5\\4&2&1\end{bmatrix}\]
Answer:
\[\det {A}=\answer {1}\]
Find \(\det {A}\) using elementary row operations.
\[A=\begin{bmatrix}3&2&2\\2&3&3\\1&1&1\end{bmatrix}\]
Answer:
\[\det {A}=\answer {0}\]
The matrix below is an elementary matrix.
(a)
What elementary row operation does this matrix perform?
(b)
Compute the determinant of the matrix in two different ways:
(i)
By cofactor expansion.
(ii)
By thinking about how the given matrix was obtained from the identity matrix.
\[E_1=\begin{bmatrix}1&0&0\\0&2&0\\0&0&1\end{bmatrix}\]
Answer:
\[\det {E_1}=\answer {2}\]
The matrix below is an elementary matrix.
(a)
What elementary row operation does this matrix perform?
(b)
Compute the determinant of the matrix in two different ways:
(i)
By cofactor expansion.
(ii)
By thinking about how the given matrix was obtained from the identity matrix.
\[E_2=\begin{bmatrix}1&0&4\\0&1&0\\0&0&1\end{bmatrix}\]
Answer:
\[\det {E_2}=\answer {1}\]
The matrix below is an elementary matrix.
(a)
What elementary row operation does this matrix perform?
(b)
Compute the determinant of the matrix in two different ways:
(i)
By cofactor expansion.
(ii)
By thinking about how the given matrix was obtained from the identity matrix.
\[E_3=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}\]
Answer:
\[\det {E_3}=\answer {-1}\]