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Section 5.1 Row Operations and Determinants (G1)

Activity 5.1.1.

The image below 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.

Figure 5.1.2. Transformation of the unit square by the matrix \(A\text{.}\)

(a)

What are the lengths of \(A\vec e_1\) and \(A\vec e_2\text{?}\)

(b)

What is the area of the transformed unit square?

Activity 5.1.3.

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.

Figure 5.1.4. Transformation of the unit square by the matrix \(B\)

(a)

What are the lengths of \(B\vec e_1\) and \(B\vec e_2\text{?}\)

(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*}
Figure 5.1.6. Certain vectors are stretched out without being rotated.

The process for finding such vectors will be covered later in this module.

Observation 5.1.7.

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 \(8\text{.}\)

Figure 5.1.8. A linear map transforming parallelograms into parallelograms.

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 5.1.9.

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.

Figure 5.1.10. The linear transformation \(B\) scaling areas by a constant factor, which we call the determinant

Activity 5.1.11.

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. What is \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\text{,}\) the area of the transformed unit square shown here?

Figure 5.1.12. The transformation of the unit square by the identity matrix.

  1. 0

  2. 1

  3. 2

  4. 4

Activity 5.1.13.

The transformation of the unit square by the standard matrix \([\vec{v}\hspace{0.5em} \vec{v}]\) is illustrated below: both \(T(\vec{e}_1)=T(\vec{e}_2)=\vec{v}\text{.}\) What is \(\det([\vec{v}\hspace{0.5em} \vec{v}])\text{,}\) the area of the transformed unit square shown here?

Figure 5.1.14. Transformation of the unit square by a matrix with identical columns.

  1. 0

  2. 1

  3. 2

  4. 4

Activity 5.1.15.

The transformations of the unit square by the standard matrices \([\vec{v}\hspace{0.5em} \vec{w}]\) and \([c\vec{v}\hspace{0.5em} \vec{w}]\) are illustrated below. Describe the value of \(\det([c\vec{v}\hspace{0.5em} \vec{w}])\text{.}\)

Figure 5.1.16. Parallelogram spanned by \(c\vec{v}\) and \(\vec{w}\)

  1. \(\displaystyle \det([\vec{v}\hspace{0.5em} \vec{w}])\)

  2. \(\displaystyle c\det([\vec{v}\hspace{0.5em} \vec{w}])\)

  3. \(\displaystyle c^2\det([\vec{v}\hspace{0.5em} \vec{w}])\)

  4. Cannot be determined from this information.

Activity 5.1.17.

The transformations of unit squares by the standard matrices \([\vec{u}\hspace{0.5em} \vec{w}]\text{,}\) \([\vec{v}\hspace{0.5em} \vec{w}]\) and \([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}]\) are illustrated below. Describe the value of \(\det([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}])\text{.}\)

Figure 5.1.18. Parallelogram spanned by \(\vec{u}+\vec{v}\) and \(\vec{w}\)

  1. \(\displaystyle \det([\vec{u}\hspace{0.5em} \vec{w}])=\det([\vec{v}\hspace{0.5em} \vec{w}])\)

  2. \(\displaystyle \det([\vec{u}\hspace{0.5em} \vec{w}])+\det([\vec{v}\hspace{0.5em} \vec{w}])\)

  3. \(\displaystyle \det([\vec{u}\hspace{0.5em} \vec{w}])\det([\vec{v}\hspace{0.5em} \vec{w}])\)

  4. Cannot be determined from this information.

Definition 5.1.19.

The determinant is the unique function \(\det:M_{n,n}\to\IR\) satisfying these properties:

  1. \(\displaystyle \det(I)=1\)

  2. \(\det(A)=0\) whenever two columns of the matrix are identical.

  3. \(\det[\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em}\cdots]= c\det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.

  4. \(\det[\cdots\hspace{0.5em}\vec{v}+\vec{w}\hspace{0.5em}\cdots]= \det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]+ \det[\cdots\hspace{0.5em}\vec{w}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.

Note that these last two properties together can be phrased as “The determinant is linear in each column.”

Observation 5.1.20.

The determinant must also satisfy other properties. Consider \(\det([\vec v \hspace{1em}\vec w+c \vec{v}])\) and \(\det([\vec v\hspace{1em}\vec w])\text{.}\)

Figure 5.1.21. Parallelogram spanned by \(\vec{w}+c\vec{v}\) and \(\vec{w}\)

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}+c\vec{w}\hspace{1em}\vec{w}]) &= \det([\vec{v}\hspace{1em}\vec{w}])+ \det([c\vec{w}\hspace{1em}\vec{w}])\\ &= \det([\vec{v}\hspace{1em}\vec{w}])+ c\det([\vec{w}\hspace{1em}\vec{w}])\\ &= \det([\vec{v}\hspace{1em}\vec{w}])+ c\cdot 0\\ &= \det([\vec{v}\hspace{1em}\vec{w}]) \end{align*}

Remark 5.1.22.

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*}
Figure 5.1.23. Reflection of a parallelogram as a result of swapping columns.

Observation 5.1.24.

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*}

Activity 5.1.26.

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?

Figure 5.1.27. Area changing under the composition of two linear maps

  1. \(\displaystyle 1\)

  2. \(\displaystyle 7\)

  3. \(\displaystyle 12\)

  4. Cannot be determined

Remark 5.1.29.

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 \(A\text{:}\) \(\left[\begin{array}{cccc} 1 & 0 & c & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]A\)

Activity 5.1.31.

Consider the row operation \(R_1+4R_3\to R_1\) applied as follows to show \(A\sim B\text{:}\)

\begin{equation*} A=\left[\begin{array}{cccc}1&2&3 & 4\\5&6 & 7 & 8\\9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16\end{array}\right] \sim \left[\begin{array}{cccc}1+4(9)&2+4(10)&3+4(11) & 4+4(12) \\5&6 & 7 & 8\\9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16\end{array}\right]=B \end{equation*}

  1. Find a matrix \(R\) such that \(B=RA\text{,}\) by applying the same row operation to \(I=\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]\text{.}\)

  2. Find \(\det R\) by comparing with the previous slide.

  3. If \(C \in M_{3,3}\) is a matrix with \(\det(C)= -3\text{,}\) find

    \begin{equation*} \det(RC)=\det(R)\det(C). \end{equation*}

Activity 5.1.32.

Consider the row operation \(R_1\leftrightarrow R_3\) applied as follows to show \(A\sim B\text{:}\)

\begin{equation*} A=\left[\begin{array}{cccc}1&2&3&4\\5&6&7&8\\9&10&11&12 \\ 13 & 14 & 15 & 16\end{array}\right] \sim \left[\begin{array}{cccc}9&10&11&12\\5&6&7&8\\1&2&3&4 \\ 13 & 14 & 15 & 16\end{array}\right]=B \end{equation*}

  1. Find a matrix \(R\) such that \(B=RA\text{,}\) by applying the same row operation to \(I\text{.}\)

  2. If \(C \in M_{3,3}\) is a matrix with \(\det(C)= 5\text{,}\) find \(\det(RC)\text{.}\)

Activity 5.1.33.

Consider the row operation \(3R_2\to R_2\) applied as follows to show \(A\sim B\text{:}\)

\begin{equation*} A=\left[\begin{array}{cccc}1&2&3&4\\5&6&7&8\\9&10&11&12 \\ 13 & 14 & 15 & 16\end{array}\right] \sim \left[\begin{array}{cccc}1&2&3&4\\3(5)&3(6)&3(7)&3(8)\\9&10&11&12 \\ 13 & 14 & 15 & 16\end{array}\right]=B \end{equation*}

  1. Find a matrix \(R\) such that \(B=RA\text{.}\)

  2. If \(C \in M_{3,3}\) is a matrix with \(\det(C)= -7\text{,}\) find \(\det(RC)\text{.}\)

Remark 5.1.34.

Recall that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant:

  1. 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*}

  2. 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*}

  3. 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 5.1.35.

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: \(\left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & c & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\)

Observation 5.1.37.

So we may compute the determinant of \(\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]\) by manipulating 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*}

Subsection 5.1.1 Videos

Figure 5.1.38. Video: Row operations, matrix multiplication, and determinants

Exercises 5.1.2 Exercises

Exercises available at checkit.clontz.org 1 .

https://checkit.clontz.org/#/banks/tbil-la/G1/