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Section 4.1 Matrices and Multiplication (MX1)

Subsection 4.1.1 Class Activities

Observation 4.1.1.

If \(T: \IR^n \rightarrow \IR^m\) and \(S: \IR^m \rightarrow \IR^k\) are linear maps, then the composition map \(S\circ T\) is a linear map from \(\IR^n \rightarrow \IR^k\text{.}\)

Figure 40. The composition of two linear maps.

Recall that for a vector, \(\vec{v} \in \IR^n\text{,}\) the composition is computed as \((S \circ T)(\vec{v})=S(T(\vec{v}))\text{.}\)

Activity 4.1.2.

Let \(T: \IR^3 \rightarrow \IR^2\) be given by the \(2\times 3\) standard matrix \(B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]\) and \(S: \IR^2 \rightarrow \IR^4\) be given by the \(4\times 2\) standard matrix \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)

What are the domain and codomain of the composition map \(S \circ T\text{?}\)

  1. The domain is \(\IR ^3\) and the codomain is \(\IR^2\)

  2. The domain is \(\IR ^2\) and the codomain is \(\IR^4\)

  3. The domain is \(\IR ^3\) and the codomain is \(\IR^4\)

  4. The domain is \(\IR ^4\) and the codomain is \(\IR^3\)

Activity 4.1.3.

Let \(T: \IR^3 \rightarrow \IR^2\) be given by the \(2\times 3\) standard matrix \(B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]\) and \(S: \IR^2 \rightarrow \IR^4\) be given by the \(4\times 2\) standard matrix \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)

What size will the standard matrix of \(S \circ T:\IR^3\to\IR^4\) be? (Rows \(\times\) Columns)

  1. \(\displaystyle 4 \times 3\)

  2. \(\displaystyle 3 \times 4\)

  3. \(\displaystyle 3 \times 2\)

  4. \(\displaystyle 2 \times 4\)

Activity 4.1.4.

Let \(T: \IR^3 \rightarrow \IR^2\) be given by the \(2\times 3\) standard matrix \(B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]\) and \(S: \IR^2 \rightarrow \IR^4\) be given by the \(4\times 2\) standard matrix \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)

(a)

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*}
(b)

Compute \((S \circ T)(\vec{e}_2) \text{.}\)

(c)

Compute \((S \circ T)(\vec{e}_3) \text{.}\)

(d)

Write the \(4\times 3\) standard matrix of \(S \circ T:\IR^3\to\IR^4\text{.}\)

Definition 4.1.5.

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 4.1.6.

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)

Find the standard matrix \(AB\) of \(S \circ T\text{.}\)

(c)

Find the standard matrix \(BA\) of \(T \circ S\text{.}\)

Activity 4.1.7.

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 4.1.8.

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 4.1.9.

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.

Answer.
\begin{equation*} AC=\left[\begin{array}{ccc} 9 & -13 & 0 \\ -9 & 12 & 2 \end{array}\right] \end{equation*}

Subsection 4.1.2 Videos

Figure 41. Video: Multiplying matrices

Subsection 4.1.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/MX1.slides.html.

Exercises 4.1.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/MX1/.

Subsection 4.1.5 Mathematical Writing Explorations

Exploration 4.1.10.

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 4.1.11.

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 4.1.12.

Use the included map in this problem.
A map with 5 dots. A is connected to B, B is connected to C, C is connected to D and E, and D and E are connected to each other
Figure 42. Adjacency map, showing roads between 5 cities

  • 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 4.1.6 Sample Problem and Solution

Sample problem Example B.1.18.