Learning Outcomes
-
Multiply matrices.
Section 4.1 Matrices and Multiplication (MX1)
Subsection 4.1.1 Warm Up
Activity 4.1.1.
Suppose that \(T\colon V\to W\) is a linear transformation.
(a)
(b)
Subsection 4.1.2 Class Activities
Observation 4.1.2.
If \(T: \IR^n \rightarrow \IR^m\) and \(S: \IR^m \rightarrow \IR^k\) are linear maps, then the composition map \(S\circ T\) computed as \((S \circ T)(\vec{v})=S(T(\vec{v}))\) is a linear map from \(\IR^n \rightarrow \IR^k\text{.}\)
Activity 4.1.3.
Let \(T: \IR^3 \rightarrow \IR^2\) be defined by the \(2\times 3\) standard matrix \(B\) and \(S: \IR^2 \rightarrow \IR^4\) be defined by the \(4\times 2\) standard matrix \(A\text{:}\)
\begin{equation*}
B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]
\hspace{2em}
A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}
\end{equation*}
(a)
What are the domain and codomain of the composition map \(S \circ T\text{?}\)
(b)
What size will the standard matrix of \(S \circ T\) be?
-
\(\displaystyle 4 \text{ (rows)} \times 3 \text{ (columns)}\)
-
\(\displaystyle 3 \text{ (rows)} \times 4 \text{ (columns)}\)
-
\(\displaystyle 3 \text{ (rows)} \times 2 \text{ (columns)}\)
-
\(\displaystyle 2 \text{ (rows)} \times 4 \text{ (columns)}\)
(c)
Compute
\begin{equation*}
(S \circ T)(\vec{e}_1)
=
S(T(\vec{e}_1))
=
S\left(\left[\begin{array}{c} 2 \\ 5\end{array}\right]\right)
=
\left[\begin{array}{c}\unknown\\\unknown\\\unknown\\\unknown\end{array}\right].
\end{equation*}
(d)
Compute \((S \circ T)(\vec{e}_2)
\text{.}\)
(e)
Compute \((S \circ T)(\vec{e}_3)
\text{.}\)
(f)
Use \((S \circ T)(\vec{e}_1),(S \circ T)(\vec{e}_2),(S \circ T)(\vec{e}_3)\) to write the standard matrix for \(S \circ T\text{.}\)
Definition 4.1.4.
We define the product \(AB\) of a \(m \times n\) matrix \(A\) and a \(n \times k\) matrix \(B\) to be the \(m \times k\) standard matrix of the composition map of the two corresponding linear functions.
For the previous activity, \(T\) was a map \(\IR^3 \rightarrow \IR^2\text{,}\) and \(S\) was a map \(\IR^2 \rightarrow \IR^4\text{,}\) so \(S \circ T\) gave a map \(\IR^3 \rightarrow \IR^4\) with a \(4\times 3\) standard matrix:
\begin{equation*}
AB
=
\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]
\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]
\end{equation*}
\begin{equation*}
=
\left[
(S \circ T)(\vec{e}_1) \hspace{1em}
(S\circ T)(\vec{e}_2) \hspace{1em}
(S \circ T)(\vec{e}_3)
\right]
=
\left[\begin{array}{ccc}
12 & -5 & 5 \\
5 & -3 & 4 \\
31 & -12 & 11 \\
-12 & 5 & -5
\end{array}\right]
.
\end{equation*}
Activity 4.1.5.
Let \(S: \IR^3 \rightarrow \IR^2\) be given by the matrix \(A=\left[\begin{array}{ccc} -4 & -2 & 3 \\ 0 & 1 & 1 \end{array}\right]\) and \(T: \IR^2 \rightarrow \IR^3\) be given by the matrix \(B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}\)
(a)
Write the dimensions (rows \(\times\) columns) for \(A\text{,}\) \(B\text{,}\) \(AB\text{,}\) and \(BA\text{.}\)
(b)
(c)
Activity 4.1.6.
Consider the following three matrices.
\begin{equation*}
A = \left[\begin{array}{ccc}1&0&-3\\3&2&1\end{array}\right]
\hspace{2em}
B = \left[\begin{array}{ccccc}2&2&1&0&1\\1&1&1&-1&0\\0&0&3&2&1\\-1&5&7&2&1\end{array}\right]
\hspace{2em}
C = \left[\begin{array}{cc}2&2\\0&-1\\3&1\\4&0\end{array}\right]
\end{equation*}
(a)
Find the domain and codomain of each of the three linear maps corresponding to \(A\text{,}\) \(B\text{,}\) and \(C\text{.}\)
(b)
Only one of the matrix products \(AB,AC,BA,BC,CA,CB\) can actually be computed. Compute it.
Activity 4.1.7.
Let \(B=\left[\begin{array}{ccc} 3 & -4 & 0 \\ 2 & 0 & -1 \\ 0 & -3 & 3 \end{array}\right]\text{,}\) and let \(A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}\)
(a)
Compute the product \(BA\) by hand.
(b)
Check your work using technology. Using Octave:
B = [3 -4 0 ; 2 0 -1 ; 0 -3 3]
A = [2 7 -1 ; 0 3 2 ; 1 1 -1]
B*A
Activity 4.1.8.
Of the following three matrices, only two may be multiplied.
\begin{equation*}
A=\left[\begin{array}{cccc}
-1 & 3 & -2 & -3 \\
1 & -4 & 2 & 3
\end{array}\right] \hspace{1em} B=\left[\begin{array}{ccc}
1 & -6 & -1 \\
0 & 1 & 0
\end{array}\right] \hspace{1em} C=\left[\begin{array}{ccc}
1 & -1 & -1 \\
0 & 1 & -2 \\
-2 & 4 & -1 \\
-2 & 3 & -1
\end{array}\right]
\end{equation*}
Explain which two can be multiplied and why. Then show how to find their product.
Activity 4.1.9.
Let \(T\left(\left[\begin{array}{c}x\\y \end{array}\right]\right)=
\left[\begin{array}{c} x+2y \\ y \\ 3x +5y \\ -x-2y \end{array}\right]\) In Fact 3.2.12 we adopted the notation
\begin{equation*}
T\left(\left[\begin{array}{c}x\\y \end{array}\right]\right)=
\left[\begin{array}{c} x+2y \\ y \\ 3x +5y \\ -x-2y \end{array}\right]=
A
\left[\begin{array}{c}x\\y \end{array}\right] =
\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]
\left[\begin{array}{c}x\\y \end{array}\right] \text{.}
\end{equation*}
Verify that \(\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]
\left[\begin{array}{c}x\\y \end{array}\right] =
\left[\begin{array}{c} x+2y \\ y \\ 3x +5y \\ -x-2y \end{array}\right]\) in terms of matrix multiplication.
Subsection 4.1.3 Individual Practice
Activity 4.1.10.
Given two \(n\times n\) matrices \(A\) and \(B\text{,}\) explain why the sentence "Multiply the matrices \(A\) and \(B \) together." is ambiguous. How could you re-write the sentence in order to eliminate the ambiguity?
Subsection 4.1.4 Videos
Subsection 4.1.5 Exercises
Exercises available at
https://library.tbil.org/2025/linear-algebra/exercises/#/bank/MX1/
Subsection 4.1.6 Mathematical Writing Explorations
Exploration 4.1.11.
Construct 3 matrices, \(A,B,\mbox{ and } C\text{,}\) such that-
\(\displaystyle AB:\mathbb{R}^4\rightarrow\mathbb{R}^2\)
-
\(\displaystyle BC:\mathbb{R}^2\rightarrow\mathbb{R}^3\)
-
\(\displaystyle CA:\mathbb{R}^3\rightarrow\mathbb{R}^4\)
-
\(\displaystyle ABC:\mathbb{R}^2\rightarrow\mathbb{R}^2\)
Exploration 4.1.12.
Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2.-
Where \(A\) and \(B\) are not square, but \(AB\) is square.
-
Where \(AB = BA\text{.}\)
-
Where \(AB \neq BA\text{.}\)
Exploration 4.1.13.
Use the included map in this problem.
-
An adjacency matrix for this map is a matrix that has the number of roads from city \(i\) to city \(j\) in the \((i,j)\) entry of the matrix. A road is a path of length exactly 1. All \((i,i)\)entries are 0. Write the adjacency matrix for this map, with the cities in alphabetical order.
-
What does the square of this matrix tell you about the map? The cube? The \(n\)-th power?
Subsection 4.1.7 Sample Problem and Solution
Sample problem Example B.1.18.
