Learning Outcomes
-
Describe how a row operation affects the determinant of a matrix.
Section 5.1 Row Operations and Determinants (GT1)
Subsection 5.1.1 Warm Up
Activity 5.1.1.
Consider the linear transformation \(T\colon \IR^2\to\IR^2\) corresponding to the standard matrix \(A=\left[\begin{matrix}1 & 3\\-1 & 2\end{matrix}\right]\text{.}\)
(a)
Draw a figure that depicts how \(T\) transforms the unit square.
(b)
What geometric features of the unit square were preserved by the transformation? Which geometric features changed?
Subsection 5.1.2 Class Activities
Observation 5.1.2.
The tool in Figure 46 can be used to visualize the effect of a linear transformation (defined by its standard matrix) on the geometry of the unit square defined by the standard basic vectors \(\vec e_1,\vec e_2\text{.}\)
Activity 5.1.3.
The image in Figure 47 illustrates how the linear transformation \(T : \IR^2 \rightarrow \IR^2\) given by the standard matrix \(A = \left[\begin{array}{cc} 2 & 0 \\ 0 & 3 \end{array}\right]\) transforms the unit square.
(a)
(b)
What is the area of the transformed unit square?
Activity 5.1.4.
The image below illustrates how the linear transformation \(S : \IR^2 \rightarrow \IR^2\) given by the standard matrix \(B = \left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\) transforms the unit square.
(a)
(b)
What is the area of the transformed unit square?
Observation 5.1.5.
It is possible to find two nonparallel vectors that are scaled but not rotated by the linear map given by \(B\text{.}\)
\begin{equation*}
B\vec e_1=\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\left[\begin{array}{c}1\\0\end{array}\right]
=\left[\begin{array}{c}2\\0\end{array}\right]=2\vec e_1
\end{equation*}
\begin{equation*}
B\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
=
\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
=
\left[\begin{array}{c}3\\2\end{array}\right]
=
4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
\end{equation*}
The process for finding such vectors will be covered later in this chapter.
Observation 5.1.6.
Notice that while a linear map can transform vectors in various ways, linear maps always transform parallelograms into parallelograms, and these areas are always transformed by the same factor: in the case of \(B=\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\text{,}\) this factor is \(8\text{.}\)
Since this change in area is always the same for a given linear map, it will be equal to the value of the transformed unit square (which begins with area \(1\)).
Remark 5.1.7.
We will define the determinant of a square matrix \(B\text{,}\) or \(\det(B)\) for short, to be the factor by which \(B\) scales areas. In order to figure out how to compute it, we first figure out the properties it must satisfy.
Activity 5.1.8.
The transformation of the unit square by the standard matrix \([\vec{e}_1\hspace{0.5em} \vec{e}_2]=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]=I\) is illustrated below. If \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\) is the area of resulting parallelogram, what is the value of \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\text{?}\)
The value for \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\) is:
-
0
-
1
-
2
-
4
Activity 5.1.9.
Which of the following is true?
-
\(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{v}])=0\)
-
\(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{v}])=1\)
-
\(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{v}])=2\)
-
\(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{v}])=4\)
Activity 5.1.10.
Which of the following is true?
-
\(\displaystyle \det([c\vec{v}\hspace{0.5em} \vec{w}])=\det([\vec{v}\hspace{0.5em} \vec{w}])\)
-
\(\displaystyle \det([c\vec{v}\hspace{0.5em} \vec{w}])=c\det([\vec{v}\hspace{0.5em} \vec{w}])\)
-
\(\displaystyle \det([c\vec{v}\hspace{0.5em} \vec{w}])=c^2\det([\vec{v}\hspace{0.5em} \vec{w}])\)
Activity 5.1.11.
Which of the following is true?
-
\(\displaystyle \det([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}])=\det([\vec{u}\hspace{0.5em} \vec{w}])=\det([\vec{v}\hspace{0.5em} \vec{w}])\)
-
\(\displaystyle \det([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}])=\det([\vec{u}\hspace{0.5em} \vec{w}])+\det([\vec{v}\hspace{0.5em} \vec{w}])\)
-
\(\displaystyle \det([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}])=\det([\vec{u}\hspace{0.5em} \vec{w}])\det([\vec{v}\hspace{0.5em} \vec{w}])\)
Definition 5.1.12.
The determinant is the unique function \(\det:M_{n,n}\to\IR\) satisfying these properties:
-
\(\displaystyle \det(I)=1\)
-
\(\det(A)=0\) whenever two columns of the matrix are identical.
-
\(\det[\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em}\cdots]= c\det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.
-
\(\det[\cdots\hspace{0.5em}\vec{v}+\vec{w}\hspace{0.5em}\cdots]= \det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]+ \det[\cdots\hspace{0.5em}\vec{w}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.
Note that these last two properties together can be phrased as “The determinant is linear in each column.”
Essentially, the determinant measures the change in “size” caused by a transformation, where “size” means area for \(2\times 2\) matrices and volume for \(3\times 3\) matrices.
Observation 5.1.13.
The determinant must also satisfy other properties. Consider \(\det([\vec v \hspace{1em}\vec w+c \vec{v}])\) and \(\det([\vec v\hspace{1em}\vec w])\text{.}\)
The base of both parallelograms is \(\vec{v}\text{,}\) while the height has not changed, so the determinant does not change either. This can also be proven using the other properties of the determinant:
\begin{align*}
\det([\vec{v}\hspace{1em}\vec{w}+c\vec{v}])
&=
\det([\vec{v}\hspace{1em}\vec{w}])+
\det([\vec{v}\hspace{1em}c\vec{v}])\\
&=
\det([\vec{v}\hspace{1em}\vec{w}])+
c\det([\vec{v}\hspace{1em}\vec{v}])\\
&=
\det([\vec{v}\hspace{1em}\vec{w}])+
c\cdot 0\\
&=
\det([\vec{v}\hspace{1em}\vec{w}])
\end{align*}
Remark 5.1.14.
Swapping columns may be thought of as a reflection, which is represented by a negative determinant. For example, the following matrices transform the unit square into the same parallelogram, but the second matrix reflects its orientation.
\begin{equation*}
A=\left[\begin{array}{cc}2&3\\0&4\end{array}\right]\hspace{1em}\det A=8\hspace{3em}
B=\left[\begin{array}{cc}3&2\\4&0\end{array}\right]\hspace{1em}\det B=-8
\end{equation*}
Observation 5.1.15.
The fact that swapping columns multiplies determinants by a negative may be verified by adding and subtracting columns.
\begin{align*}
\det([\vec{v}\hspace{1em}\vec{w}])
&=
\det([\vec{v}+\vec{w}\hspace{1em}\vec{w}])\\
&=
\det([\vec{v}+\vec{w}\hspace{1em}\vec{w}-(\vec{v}+\vec{w})])\\
&=
\det([\vec{v}+\vec{w}\hspace{1em}-\vec{v}])\\
&=
\det([\vec{v}+\vec{w}-\vec{v}\hspace{1em}-\vec{v}])\\
&=
\det([\vec{w}\hspace{1em}-\vec{v}])\\
&=
-\det([\vec{w}\hspace{1em}\vec{v}])
\end{align*}
Fact 5.1.16.
To summarize, we’ve shown that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant in the following way:
-
Multiplying a column by a scalar multiplies the determinant by that scalar:\begin{equation*} c\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots])= \det([\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em} \cdots]) \end{equation*}
-
Swapping two columns changes the sign of the determinant:\begin{equation*} \det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])= -\det([\cdots\hspace{0.5em}\vec{w}\hspace{0.5em} \cdots\hspace{1em}\vec{v}\hspace{0.5em} \cdots]) \end{equation*}
-
Adding a multiple of a column to another column does not change the determinant:\begin{equation*} \det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])= \det([\cdots\hspace{0.5em}\vec{v}+c\vec{w}\hspace{0.5em} \cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots]) \end{equation*}
Activity 5.1.17.
The transformation given by the standard matrix \(A\) scales areas by \(4\text{,}\) and the transformation given by the standard matrix \(B\) scales areas by \(3\text{.}\) By what factor does the transformation given by the standard matrix \(AB\) scale areas?
-
\(\displaystyle 1\)
-
\(\displaystyle 7\)
-
\(\displaystyle 12\)
-
Cannot be determined
Fact 5.1.18.
Since the transformation given by the standard matrix \(AB\) is obtained by applying the transformations given by \(A\) and \(B\text{,}\) it follows that
\begin{equation*}
\det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA)\text{.}
\end{equation*}
Remark 5.1.19.
Recall that row operations may be produced by matrix multiplication.
-
Multiply the first row of \(A\) by \(c\text{:}\) \(\left[\begin{array}{cccc} c & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]A\)
-
Swap the first and second row of \(A\text{:}\) \(\left[\begin{array}{cccc} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]A\)
-
Add \(c\) times the third row to the first row of \(A\text{:}\) \(\left[\begin{array}{cccc} 1 & 0 & c & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]A\)
Fact 5.1.20.
The determinants of row operation matrices may be computed by manipulating columns to reduce each matrix to the identity:
-
Scaling a row: \(\det \left[\begin{array}{cccc} c & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] = c\det \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] = c\)
-
Swapping rows: \(\det \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] = -1\det \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] = -1\)
-
Adding a row multiple to another row: \(\det \left[\begin{array}{cccc} 1 & 0 & c & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] = \det \left[\begin{array}{cccc} 1 & 0 & c-1c & 0\\ 0 & 1 & 0-0c & 0\\ 0 & 0 & 1-0c & 0 \\ 0 & 0 & 0-0c & 1 \end{array}\right] = \det(I)=1\)
Activity 5.1.21.
\begin{equation*}
A=\left[\begin{array}{cccc}1&2&0&-3\\4&1&3&0\\0&0&-3&-5\\1&1&1&3\end{array}\right]
\sim
\left[\begin{array}{cccc}1+4(0)&2+4(0)&0+4(-3)&-3+4(-5)\\4&1&3&0\\0&0&-3&-5\\1&1&1&3\end{array}\right]=B
\end{equation*}
(a)
Find a matrix \(R\) such that \(B=RA\text{,}\) by applying the same row operation to \(I=\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]\text{.}\)
(b)
The determinant of \(A\) is \(70\text{.}\) Complete the following computation to calculate the determinant of \(B\text{:}\)
\begin{align*}
\det(B) &= \det(RA)\\
&= \det(R)\det(A)\\
&= (\unknown)(\unknown)\\
&= \unknown
\end{align*}
Activity 5.1.22.
Consider the row operation \(R_1\leftrightarrow R_4\) applied as follows to show \(A\sim B\text{:}\)
\begin{equation*}
A=\left[\begin{array}{cccc}1&2&0&-3\\4&1&3&0\\0&0&-3&-5\\1&1&1&3\end{array}\right]
\sim
\left[\begin{array}{cccc}1&1&1&3\\4&1&3&0\\0&0&-3&-5\\1&2&0&-3\end{array}\right]=B
\end{equation*}
(a)
(b)
Activity 5.1.23.
\begin{equation*}
A=\left[\begin{array}{cccc}1&2&0&-3\\4&1&3&0\\0&0&-3&-5\\1&1&1&3\end{array}\right]
\sim
\left[\begin{array}{cccc}1&2&0&-3\\3(4)&3(1)&3(3)&3(0)\\0&0&-3&-5\\1&1&1&3\end{array}\right] =B
\end{equation*}
(a)
(b)
Activity 5.1.24.
(a)
Let \(B\) be the matrix obtained from \(A\) by applying the row operation \(R_1 - 5 R_3 \to R_1\text{.}\) What is \(\mathrm{det}\,B\text{?}\)
(b)
Let \(M\) be the matrix obtained from \(A\) by applying the row operation \(R_3 \leftrightarrow R_1\text{.}\) What is \(\mathrm{det}\,M\text{?}\)
(c)
Let \(P\) be the matrix obtained from \(A\) by applying the row operation \(2 R_4 \to R_4\text{.}\) What is \(\mathrm{det}\,P\text{?}\)
Remark 5.1.25.
Recall that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant:
-
Multiplying columns by scalars:\begin{equation*} \det([\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em} \cdots])= c\det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots]) \end{equation*}
-
Swapping two columns:\begin{equation*} \det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])= -\det([\cdots\hspace{0.5em}\vec{w}\hspace{0.5em} \cdots\hspace{1em}\vec{v}\hspace{0.5em} \cdots]) \end{equation*}
-
Adding a multiple of a column to another column:\begin{equation*} \det([\cdots\hspace{0.5em}\vec{v}\hspace{0.5em} \cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots])= \det([\cdots\hspace{0.5em}\vec{v}+c\vec{w}\hspace{0.5em} \cdots\hspace{1em}\vec{w}\hspace{0.5em} \cdots]) \end{equation*}
Remark 5.1.26.
The determinants of row operation matrices may be computed by manipulating columns to reduce each matrix to the identity:
-
Scaling a row: \(\left[\begin{array}{cccc} 1 & 0 & 0 &0 \\ 0 & c & 0 &0\\ 0 & 0 & 1 &0 \\ 0 & 0 & 0 & 0 \end{array}\right]\)
-
Swapping rows: \(\left[\begin{array}{cccc} 0 & 1 & 0 &0 \\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\)
-
Adding a row multiple to another row: \(\left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & c & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\)
Fact 5.1.27.
Thus we can also use both row operations to simplify determinants:
-
Multiplying rows by scalars:\begin{equation*} \det\left[\begin{array}{c}\vdots\\cR\\\vdots\end{array}\right]= c\det\left[\begin{array}{c}\vdots\\R\\\vdots\end{array}\right] \end{equation*}
-
Swapping two rows:\begin{equation*} \det\left[\begin{array}{c}\vdots\\R\\\vdots\\S\\\vdots\end{array}\right]= -\det\left[\begin{array}{c}\vdots\\S\\\vdots\\R\\\vdots\end{array}\right] \end{equation*}
-
Adding multiples of rows/columns to other rows:\begin{equation*} \det\left[\begin{array}{c}\vdots\\R\\\vdots\\S\\\vdots\end{array}\right]= \det\left[\begin{array}{c}\vdots\\R+cS\\\vdots\\S\\\vdots\end{array}\right] \end{equation*}
Activity 5.1.28.
Complete the following derivation for a formula calculating \(2\times 2\) determinants:
\begin{align*}
\det\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]
&=
\unknown \det \left[\begin{array}{cc} 1 & b/a \\ c & d \end{array}\right]\\
&=
\unknown \det \left[\begin{array}{cc} 1 & b/a \\ c-c & d-bc/a \end{array}\right]\\
&=
\unknown \det \left[\begin{array}{cc} 1 & b/a \\ 0 & d-bc/a \end{array}\right]\\
&=
\unknown \det \left[\begin{array}{cc} 1 & b/a \\ 0 & 1 \end{array}\right]\\
&=
\unknown \det \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\\
&=
\unknown \det I\\
&=
\unknown
\end{align*}
Observation 5.1.29.
So we may compute the determinant of \(\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]\) by using determinant properties to manipulate its rows/columns to reduce the matrix to \(I\text{:}\)
\begin{align*}
\det\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]
&=
2 \det \left[\begin{array}{cc} 1 & 2 \\ 2 & 3 \end{array}\right]\\
&=
%2 \det \left[\begin{array}{cc} 1 & 2 \\ 2-2(1) & 3-2(2)\end{array}\right]=
2 \det \left[\begin{array}{cc} 1 & 2 \\ 0 & -1 \end{array}\right]\\
&=
%2(-1) \det \left[\begin{array}{cc} 1 & -2 \\ 0 & +1 \end{array}\right]=
-2 \det \left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \end{array}\right]\\
&=
%-2 \det \left[\begin{array}{cc} 1+2(0) & -2+2(1) \\ 0 & 1\end{array}\right] =
-2 \det \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]\\
&=
%-2\det I =
%-2(1) =
-2
\end{align*}
Or we may use a formula:
\begin{equation*}
\det\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]
=
(2)(3)-(4)(2)=-2
\end{equation*}
Subsection 5.1.3 Individual Practice
Activity 5.1.30.
Suppose we have a linear transformation \(T\colon\IR^2\to\IR^2\text{.}\) Given some shape \(S\) in the plane \(\IR^2\text{,}\) we can use to \(T\) to transform it into some new shape \(T(S)\text{.}\) Consider the following questions about properties that may or may not be preserved by \(T\text{.}\)
(a)
(b)
If \(S\) is a straight line segment, does \(T(S)\) necessarily have to have the same length as that of \(S\text{?}\)
(c)
(d)
Continuing as above, do the angles of \(T(S)\) necessarily have to be the same as those of \(S\text{?}\)
Subsection 5.1.4 Videos
Subsection 5.1.5 Exercises
Exercises available at
https://library.tbil.org/2025/linear-algebra/exercises/#/bank/GT1/
Subsection 5.1.6 Mathematical Writing Explorations
Exploration 5.1.31.
-
Prove or disprove. The determinant is a linear operator on the vector space of \(n \times n\) matrices.
-
Find a matrix that will double the area of a region in \(\mathbb{R}^2\text{.}\)
-
Find a matrix that will triple the area of a region in \(\mathbb{R}^2\text{.}\)
-
Find a matrix that will halve the area of a region in \(\mathbb{R}^2\text{.}\)
Subsection 5.1.7 Sample Problem and Solution
Sample problem Example B.1.22.
