Learning Outcomes
-
Compute the determinant of a \(4\times 4\) matrix.
Section 5.2 Computing Determinants (GT2)
Subsection 5.2.1 Warm Up
Activity 5.2.1.
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 5.2.2 Class Activities
Remark 5.2.2.
We’ve seen that row reducing all the way into RREF gives us a method of computing determinants.
However, we learned in Chapter 1 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 5.2.3.
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 5.2.4.
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 5.2.5.
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 5.2.6.
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 5.2.7.
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 5.2.8.
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 5.2.9.
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 5.2.10.
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 5.2.11.
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 5.2.12.
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 5.2.13.
Recall the formula for a \(2\times 2\) determinant found in Observation 5.1.29:
\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 5.2.14.
Based on the previous activities, which technique is easier for computing determinants?
-
Memorizing formulas.
-
Using row/column operations.
-
Laplace expansion.
-
Some other technique.
Activity 5.2.15.
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 5.2.16.
Subsection 5.2.3 Individual Practice
Activity 5.2.17.
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 5.2.4 Videos
Subsection 5.2.5 Exercises
Exercises available at
https://library.tbil.org/2025/linear-algebra/exercises/#/bank/GT2/
Subsection 5.2.6 Mathematical Writing Explorations
Exploration 5.2.18.
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 5.2.19.
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 5.2.20.
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 5.2.7 Sample Problem and Solution
Sample problem Example B.1.23.
