TBIL-LA Matrices and Multiplication (MX1)

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Section 4.1 Matrices and Multiplication (MX1)

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Learning Outcomes

Multiply matrices.

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Subsection 4.1.1 Class Activities

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Observation 4.1.1.

If $T : R^{n} \to R^{m}$ and $S : R^{m} \to R^{k}$ are linear maps, then the composition map $S \circ T$ is a linear map from $R^{n} \to R^{k} .$

Figure 40. The composition of two linear maps.

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

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Activity 4.1.2.

Let $T : R^{3} \to R^{2}$ be given by the $2 \times 3$ standard matrix $B = [\begin{array}{ccc} 2 & 1 & - 3 \\ 5 & - 3 & 4 \end{array}]$ and $S : R^{2} \to R^{4}$ be given by the $4 \times 2$ standard matrix $A = [\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ - 1 & - 2 \end{array}] .$

What are the domain and codomain of the composition map $S \circ T ?$

The domain is $R^{3}$ and the codomain is $R^{2}$
The domain is $R^{2}$ and the codomain is $R^{4}$
The domain is $R^{3}$ and the codomain is $R^{4}$
The domain is $R^{4}$ and the codomain is $R^{3}$

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Activity 4.1.3.

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

$4 \times 3$
$3 \times 4$
$3 \times 2$
$2 \times 4$

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Activity 4.1.4.

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(a)

Compute

(S \circ T) ({\vec{e}}_{1}) = S (T ({\vec{e}}_{1})) = S ([\begin{matrix} 2 \\ 5 \end{matrix}]) = [\begin{matrix} ? \\ ? \\ ? \\ ? \end{matrix}] .

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(b)

Compute $(S \circ T) ({\vec{e}}_{2}) .$

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(c)

Compute $(S \circ T) ({\vec{e}}_{3}) .$

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(d)

Write the $4 \times 3$ standard matrix of $S \circ T : R^{3} \to R^{4} .$

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Definition 4.1.5.

We define the product $A B$ 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 $R^{3} \to R^{2},$ and $S$ was a map $R^{2} \to R^{4},$ so $S \circ T$ gave a map $R^{3} \to R^{4}$ with a $4 \times 3$ standard matrix:

A B = [\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ - 1 & - 2 \end{array}] [\begin{array}{ccc} 2 & 1 & - 3 \\ 5 & - 3 & 4 \end{array}]

= [(S \circ T) ({\vec{e}}_{1}) (S \circ T) ({\vec{e}}_{2}) (S \circ T) ({\vec{e}}_{3})] = [\begin{array}{ccc} 12 & - 5 & 5 \\ 5 & - 3 & 4 \\ 31 & - 12 & 11 \\ - 12 & 5 & - 5 \end{array}] .

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Activity 4.1.6.

Let $S : R^{3} \to R^{2}$ be given by the matrix $A = [\begin{array}{ccc} - 4 & - 2 & 3 \\ 0 & 1 & 1 \end{array}]$ and $T : R^{2} \to R^{3}$ be given by the matrix $B = [\begin{array}{cc} 2 & 3 \\ 1 & - 1 \\ 0 & - 1 \end{array}] .$

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(a)

Write the dimensions (rows $\times$ columns) for $A,$ $B,$ $A B,$ and $B A .$

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(b)

Find the standard matrix $A B$ of $S \circ T .$

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(c)

Find the standard matrix $B A$ of $T \circ S .$

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Activity 4.1.7.

Consider the following three matrices.

A = [\begin{array}{ccc} 1 & 0 & - 3 \\ 3 & 2 & 1 \end{array}] B = [\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}] C = [\begin{array}{cc} 2 & 2 \\ 0 & - 1 \\ 3 & 1 \\ 4 & 0 \end{array}]

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(a)

Find the domain and codomain of each of the three linear maps corresponding to $A,$ $B,$ and $C .$

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(b)

Only one of the matrix products $A B, A C, B A, B C, C A, C B$ can actually be computed. Compute it.

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Activity 4.1.8.

Let $B = [\begin{array}{ccc} 3 & - 4 & 0 \\ 2 & 0 & - 1 \\ 0 & - 3 & 3 \end{array}],$ and let $A = [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] .$

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(a)

Compute the product $B A$ by hand.

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(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


    
        
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B = [3 -4 0 ; 2 0 -1 ; 0 -3 3]
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A = [2 7 -1 ; 0 3 2  ; 1 1 -1]
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B*A

    
    
    
    
        
            
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Activity 4.1.9.

Of the following three matrices, only two may be multiplied.

A = [\begin{array}{cccc} - 1 & 3 & - 2 & - 3 \\ 1 & - 4 & 2 & 3 \end{array}] B = [\begin{array}{ccc} 1 & - 6 & - 1 \\ 0 & 1 & 0 \end{array}] C = [\begin{array}{ccc} 1 & - 1 & - 1 \\ 0 & 1 & - 2 \\ - 2 & 4 & - 1 \\ - 2 & 3 & - 1 \end{array}]

Explain which two can be multiplied and why. Then show how to find their product.

Answer.

A C = [\begin{array}{ccc} 9 & - 13 & 0 \\ - 9 & 12 & 2 \end{array}]

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Subsection 4.1.2 Videos

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Figure 41. Video: Multiplying matrices

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Exploration 4.1.10.

Construct 3 matrices,

A, B, and C,

such that

$A B : R^{4} \to R^{2}$
$B C : R^{2} \to R^{3}$
$C A : R^{3} \to R^{4}$
$A B C : R^{2} \to R^{2}$

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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 $A B$ is square.
Where $A B = B A .$
Where $A B \neq B A .$

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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?

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Subsection 4.1.6 Sample Problem and Solution

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Sample problem Example B.1.18.