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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 calculuate 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 \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 0 & 1\end{array}\right]\text{.}\)
Figure 56. Transformation of the unit cube by the linear transformation.
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?
  1. \(\displaystyle \det \left[\begin{array}{cc} 1 & 1 \\ 1 & 3 \end{array}\right]\)
  2. \(\displaystyle \det \left[\begin{array}{cc} 1 & 0 \\ 3 & 1 \end{array}\right]\)
  3. \(\displaystyle \det \left[\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right]\)
  4. \(\displaystyle \det \left[\begin{array}{cc} 1 & 3 \\ 0 & 0 \end{array}\right]\)

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 5.1.28:
\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?
  1. Memorizing formulas.
  2. Using row/column operations.
  3. Laplace expansion.
  4. 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.

You can check your answers using technology.

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

Figure 57. Video: Simplifying a determinant using row operations
Figure 58. Video: Computing a determinant

Exercises 5.2.5 Exercises

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.