Processing math: 100%
Skip to main content
🔗

Section 9.3 Examples

🔗

Subsection 9.3.1 Determinants by row reduction

🔗
As discussed in Warning 8.3.3, determinants by cofactor expansions are extremely inefficient for matrices larger than 3×3. Here we provide an example of using the row reduction method to compute a determinant.

🔗
Example 9.3.1. Using row reduction to compute a determinant.

Let's recompute the determinant of

A=[1−1212011010−31−2−10],

the same matrix from Example 8.4.5.

First, let's row reduce. For the purposes of describing our thinking in using the matrix reduction calculation to determine the determinant of A, we'll label our matrices as we go.

A=[1−1212011010−31−2−10]xR2−2R1xR4−R1⟶A1=[1−12102−3−1010−30−1−3−1]xR2↔R3x⟶A2=[1−121010−302−3−10−1−3−1]xxR3−2R2R4+R2⟶A3=[1−121010−300−3500−3−4]xx−13R3x⟶A4=[1−121010−3001−5300−3−4]xxxR4+3R3⟶A5=[1−121010−3001−53000−9]

We would need one more operation to get to REF, but we are already at upper triangular so we don't need to bother. And notice that we didn't bother clearing entries above leading ones, since our goal was to get to an upper triangular matrix, which only requires entries below leading ones to be cleared.

Now we'll work backwards to determine detA.

A5

This last matrix is upper triangular, so its determinant is equal to the product of the diagonal entries: detA5=1⋅1⋅1⋅(−9)=−9.

A4

Matrix A5 was produced from A4 by an operation that does not change the determinant, so detA4 must be −9 as well.

A3

Matrix A4 was produced from A3 by multiplying a row, so detA4=−13detA3. Solving for detA3, we get detA3=−3⋅(−9)=27.

A2

Matrix A3 was produced from A2 by an operation that does not change the determinant, so detA2 must be 27 as well.

A1

Matrix A2 was produced from A1 by swapping rows, so these two determinants have opposite signs. Thus, detA1=−27.

A

Matrix A1 was produced from A by a pair of operations, neither of which changes the determinant, so finally we have detA=−27.

This analysis agrees with the calculation of detA by cofactor expansion in Example 8.4.5.

🔗

Subsection 9.3.2 Matrices of determinant zero

🔗
Example 9.3.2. Recognizing detA=0.

Here are a few examples of recognizing matrices that have determinant 0.

  1. [1−121010−3010−31234]
  2. [1−1210155710−3−22−4−2]
  3. [1−101410−3−110−31204]
  4. [1−12−101517101−22−42]

The first matrix has two identical rows, the second matrix has two proportional rows (R4=−2R1), the third matrix has a column of zeros, and the fourth matrix has two identical columns. So the determinant of each of these matrices is 0.