Learning Outcomes
- Compute the determinant of a \(4\times 4\) matrix.
Section 22.2 Computing Determinants (GT2)
Subsection 22.2.1 Warm Up
Activity 22.32.
Consider the matrix \(A=\left[\begin{matrix}1 & 2\\ 3 & 4\end{matrix}\right]\text{.}\)
(a)
Use a combination of row and column operations to transform \(A\) into the identity matrix. Use this to calculate the determinant of \(A\text{.}\)
(b)
Check your work using the formula for the determinant of a \(2\times 2\) matrix.
Subsection 22.2.2 Class Activities
Remark 22.33.
We’ve seen that row reducing all the way into RREF gives us a method of computing determinants.
However, we learned in Chapter 18 that this can be tedious for large matrices. Thus, we will try to figure out how to turn the determinant of a larger matrix into the determinant of a smaller matrix.
Activity 22.34.
The following image illustrates the transformation of the unit cube by the matrix \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 0 & 1\end{array}\right]\text{.}\)
Recall that for this solid \(V=Bh\text{,}\) where \(h\) is the height of the solid and \(B\) is the area of its parallelogram base. So what must its volume be?
- \(\displaystyle \det \left[\begin{array}{cc} 1 & 1 \\ 1 & 3 \end{array}\right]\)
- \(\displaystyle \det \left[\begin{array}{cc} 1 & 0 \\ 3 & 1 \end{array}\right]\)
- \(\displaystyle \det \left[\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right]\)
- \(\displaystyle \det \left[\begin{array}{cc} 1 & 3 \\ 0 & 0 \end{array}\right]\)
Fact 22.35.
If row \(i\) contains all zeros except for a \(1\) on the main (upper-left to lower-right) diagonal, then both column and row \(i\) may be removed without changing the value of the determinant.
\begin{equation*}
\det \left[\begin{array}{cccc}
3 & {\color{red} 2} & -1 & 3 \\
{\color{red} 0} & {\color{red} 1}
& {\color{red} 0} & {\color{red} 0} \\
-1 & {\color{red} 4} & 1 & 0 \\
5 & {\color{red} 0} & 11 & 1
\end{array}\right] =
\det \left[\begin{array}{ccc}
3 & -1 & 3 \\
-1 & 1 & 0 \\
5 & 11 & 1
\end{array}\right]
\end{equation*}
Since row and column operations affect the determinants in the same way, the same technique works for a column of all zeros except for a \(1\) on the main diagonal.
\begin{equation*}
\det \left[\begin{array}{cccc}
3 & {\color{red} 0} & -1 & 5 \\
{\color{red} 2} & {\color{red} 1} & {\color{red} 4} &
{\color{red} 0} \\
-1 & {\color{red} 0} & 1 & 11 \\
3 & {\color{red} 0} & 0 & 1
\end{array}\right] =
\det \left[\begin{array}{ccc}
3 & -1 & 5 \\
-1 & 1 & 11 \\
3 & 0 & 1
\end{array}\right]
\end{equation*}
Put another way, if you have either a column or row from the identity matrix, you can cancel both the column and row containing the \(1\text{.}\)
Warning 22.36.
If the \(1\) is not on the main diagonal, you’ll need to use row or column swaps in order to cancel.
\begin{equation*}
\det \left[\begin{array}{cccc}
3 & {\color{red} 0} & -1 & 5 \\
-1 & {\color{red} 0} & 1 & 11 \\
{\color{red} 2} & {\color{red} 1} & {\color{red} 4} &
{\color{red} 0} \\
3 & {\color{red} 0} & 0 & 1
\end{array}\right] =
-\det \left[\begin{array}{cccc}
3 & {\color{red} 0} & -1 & 5 \\
{\color{red} 2} & {\color{red} 1} & {\color{red} 4} &
{\color{red} 0} \\
-1 & {\color{red} 0} & 1 & 11 \\
3 & {\color{red} 0} & 0 & 1
\end{array}\right] =
-\det \left[\begin{array}{ccc}
3 & -1 & 5 \\
-1 & 1 & 11 \\
3 & 0 & 1
\end{array}\right]
\end{equation*}
Activity 22.37.
Remove an appropriate row and column of \(\det \left[\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 5 & 12 \\ 3 & 2 & -1 \end{array}\right]\) to simplify the determinant to a \(2\times 2\) determinant.
Activity 22.38.
Simplify \(\det \left[\begin{array}{ccc} 0 & 3 & -2 \\ 2 & 5 & 12 \\ 0 & 2 & -1 \end{array}\right]\) to a multiple of a \(2\times 2\) determinant by first doing the following:
(a)
Factor out a \(2\) from a column.
(b)
Swap rows or columns to put a \(1\) on the main diagonal.
Activity 22.39.
Simplify \(\det \left[\begin{array}{ccc} 4 & -2 & 2 \\ 3 & 1 & 4 \\ 1 & -1 & 3\end{array}\right]\) to a multiple of a \(2\times 2\) determinant by first doing the following:
(a)
Use row/column operations to create two zeroes in the same row or column.
(b)
Factor/swap as needed to get a row/column of all zeroes except a \(1\) on the main diagonal.
Observation 22.40.
Using row/column operations, you can introduce zeros and reduce dimension to whittle down the determinant of a large matrix to a determinant of a smaller matrix.
\begin{align*}
\det\left[\begin{array}{cccc}
4 & 3 & 0 & 1 \\
2 & -2 & 4 & 0 \\
-1 & 4 & 1 & 5 \\
2 & 8 & 0 & 3
\end{array}\right]
&=
\det\left[\begin{array}{cccc}
4 & 3 & {\color{red} 0} & 1 \\
6 & -18 & {\color{red} 0} & -20 \\
{\color{red} {-1}} & {\color{red} 4} &
{\color{red} 1} & {\color{red} 5} \\
2 & 8 & {\color{red} 0} & 3
\end{array}\right]
=
\det\left[\begin{array}{ccc}
4 & 3 & 1 \\
6 & -18 & -20 \\
2 & 8 & 3
\end{array}\right]\\
&=\dots=
-2\det\left[\begin{array}{ccc}
{\color{red} 1} & {\color{red} 3} & {\color{red} 4} \\
{\color{red} 0} & 21 & 43 \\
{\color{red} 0} & -1 & -10
\end{array}\right]
=
-2\det\left[\begin{array}{cc} 21 & 43 \\ -1 & -10 \end{array}\right]\\
&= \dots=
-2\det\left[\begin{array}{cc}
-167 & {\color{red}{21}} \\
{\color{red} 0} & {\color{red} 1}
\end{array}\right]
= -2\det[-167]\\
&=-2(-167)\det(I)=
334
\end{align*}
Activity 22.41.
Rewrite
\begin{equation*}
\det \left[\begin{array}{cccc} 2 & 1 & -2 & 1 \\ 3 & 0 & 1 & 4
\\ -2 & 2 & 3 & 0 \\ -2 & 0 & -3 & -3 \end{array}\right]
\end{equation*}
as a multiple of a determinant of a \(3\times3\) matrix.
Activity 22.42.
Compute \(\det\left[\begin{array}{cccc}
2 & 3 & 5 & 0 \\
0 & 3 & 2 & 0 \\
1 & 2 & 0 & 3 \\
-1 & -1 & 2 & 2
\end{array}\right]\) by using any combination of row/column operations.
Observation 22.43.
Another option is to take advantage of the fact that the determinant is linear in each row or column. This approach is called Laplace expansion or cofactor expansion.
For example, since \(\color{blue}{
\left[\begin{array}{ccc} 1 & 2 & 4 \end{array}\right]
=
1\left[\begin{array}{ccc} 1 & 0 & 0 \end{array}\right]
+
2\left[\begin{array}{ccc} 0 & 1 & 0 \end{array}\right]
+
4\left[\begin{array}{ccc} 0 & 0 & 1 \end{array}\right]}
\text{,}\)
\begin{align*}
\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 1} & {\color{blue} 2} & {\color{blue} 4} \end{array}\right] &=
{\color{blue} 1}\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 1} & {\color{blue} 0} & {\color{blue} 0} \end{array}\right] +
{\color{blue} 2}\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 0} & {\color{blue} 1} & {\color{blue} 0} \end{array}\right] +
{\color{blue} 4}\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ {\color{blue} 0} & {\color{blue} 0} & {\color{blue} 1} \end{array}\right]\\
&= -1\det \left[\begin{array}{ccc} 5 & 3 & 2 \\ 5 & 3 & -1 \\ 0 & 0 & 1 \end{array}\right]
-2\det \left[\begin{array}{ccc} 2 & 5 & 3 \\ -1 & 5 & 3 \\ 0 & 0 & 1 \end{array}\right] +
4\det \left[\begin{array}{ccc} 2 & 3 & 5 \\ -1 & 3 & 5 \\ 0 & 0 & 1 \end{array}\right]\\
&= -\det \left[\begin{array}{cc} 5 & 3 \\ 5 & 3 \end{array}\right]
-2 \det \left[\begin{array}{cc} 2 & 5 \\ -1 & 5 \end{array}\right]
+4 \det \left[\begin{array}{cc} 2 & 3 \\ -1 & 3 \end{array}\right]
\end{align*}
Observation 22.44.
Recall the formula for a \(2\times 2\) determinant found in Observation 22.29:
\begin{equation*}
\det \left[\begin{array}{cc} a & b \\ c & d \end{array}\right]
=
ad-bc\text{.}
\end{equation*}
There are formulas and algorithms for the determinants of larger matrices, but they can be pretty tedious to use. For example, writing out a formula for a \(4\times 4\) determinant would require 24 different terms!
\begin{equation*}
\det\left[\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array}\right]
=
a_{11}(a_{22}(a_{33}a_{44}-a_{43}a_{34})-a_{23}(a_{32}a_{44}-a_{42}a_{34})+\dots)+\dots
\end{equation*}
Activity 22.45.
Based on the previous activities, which technique is easier for computing determinants?
- Memorizing formulas.
- Using row/column operations.
- Laplace expansion.
- Some other technique.
Activity 22.46.
Use your preferred technique to compute \(\det\left[\begin{array}{cccc}
4 & -3 & 0 & 0 \\
1 & -3 & 2 & -1 \\
3 & 2 & 0 & 3 \\
0 & -3 & 2 & -2
\end{array}\right]
\text{.}\)
Insight 22.47.
Subsection 22.2.3 Individual Practice
Activity 22.48.
A diagonal matrix is a matrix that has zeroes in every position except (possibly) the main upper-left to lower-right diagonal. A matrix is upper (resp. lower) triangular if every entry below (resp. above) the main diagonal is zero.
(a)
Explain why the determinant of a diagonal matrix is always equal to the product of the entries on the main diagonal.
(b)
Explain why the determinant of an upper (or lower) triangular matrix is always equal to the product of the entries on the main diagonal.
Subsection 22.2.4 Videos
Exercises 22.2.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/GT2/.Subsection 22.2.6 Mathematical Writing Explorations
Exploration 22.49.
Prove that the equation of a line in the plane, through points \((x_1,y_1), (x_2,y_2)\text{,}\) when \(x_1 \neq x_2\) is given by the equation \(\mbox{det}\left(\begin{array}{ccc}x&y&1\\x_1&y_1&1\\x_2&y_2&1\end{array}\right) = 0.\)Exploration 22.50.
Show that, if an \(n \times n\) matrix \(M\) has a non-zero determinant, then any \(\vec{v} \in \mathbb{R}^n\) can be represented as a linear combination of the columns of \(M\text{.}\)Exploration 22.51.
What is the smallest number of zeros necessary to place in a \(4 \times 4\) matrix, and the placement of those zeros, such that the matrix has a zero determinant?Subsection 22.2.7 Sample Problem and Solution
Sample problem Example C.23.
