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Section 4.1 Matrices and Multiplication (M1)

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 \(\IR^n \rightarrow \IR^k\text{.}\)

\begin{tikzcd}[ampersand replacement=\&]
\IR^n \arrow[rr, bend right, "S\circ T"']  \arrow[r,"T"] \& \IR^m  \arrow[r,"S"]  \&\IR^k 
\end{tikzcd}

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 4.1.1.

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

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

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

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

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

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

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

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

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

  1. \(\displaystyle 4 \times 3\)
  2. \(\displaystyle 4 \times 2\)
  3. \(\displaystyle 3 \times 4\)
  4. \(\displaystyle 3 \times 2\)
  5. \(\displaystyle 2 \times 4\)
  6. \(\displaystyle 2 \times 3\)
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 \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)

(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 4.1.2.

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 4.1.4.

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 4.1.5.

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 4.1.6.

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

Exercises 4.1.1 Exercises

1.

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

\begin{align*} A= \left[\begin{array}{cc} -1 & 4 \\ 4 & 5 \\ -2 & 3 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & 6 & -2 \\ -1 & 4 & 5 \\ 0 & 3 & 1 \\ 1 & 1 & -2 \end{array}\right] & & C= \left[\begin{array}{cc} 3 & -2 \\ 2 & -1 \\ -1 & 0 \\ 0 & -4 \end{array}\right] \end{align*}

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

Answer
\begin{equation*} BA= \left[\begin{array}{cc} 27 & 28 \\ 7 & 31 \\ 10 & 18 \\ 7 & 3 \end{array}\right] \end{equation*}
2.

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

\begin{align*} A= \left[\begin{array}{cccc} 1 & 4 & -1 & 0 \\ -1 & -4 & 2 & -1 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & -1 & -4 \\ 2 & -1 & -5 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & -5 & 5 & 5 \\ 1 & -4 & 3 & 3 \\ 1 & -5 & 6 & 6 \end{array}\right] \end{align*}

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

Answer
\begin{equation*} BC= \left[\begin{array}{cccc} -4 & 19 & -22 & -22 \\ -4 & 19 & -23 & -23 \end{array}\right] \end{equation*}
3.

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

\begin{align*} A= \left[\begin{array}{cccc} 1 & 1 & -3 & -2 \\ 0 & 1 & -4 & -1 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ -2 & -1 & -1 & -1 \\ -1 & 3 & -3 & 5 \end{array}\right] & & C= \left[\begin{array}{cc} 1 & -3 \\ -1 & 4 \\ -1 & 4 \end{array}\right] \end{align*}

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

Answer
\begin{equation*} CA= \left[\begin{array}{cccc} 1 & -2 & 9 & 1 \\ -1 & 3 & -13 & -2 \\ -1 & 3 & -13 & -2 \end{array}\right] \end{equation*}
4.

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

\begin{align*} A= \left[\begin{array}{cccc} 1 & -3 & 4 & 6 \\ 0 & 1 & -2 & -1 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & 5 & 6 \\ 0 & 0 & 1 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & -1 & -5 \\ 0 & 1 & 3 \\ 1 & -1 & -4 \\ 0 & 0 & 2 \end{array}\right] \end{align*}

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

Answer
\begin{equation*} AC= \left[\begin{array}{ccc} 5 & -8 & -18 \\ -2 & 3 & 9 \end{array}\right] \end{equation*}
5.

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

\begin{align*} A= \left[\begin{array}{ccc} 2 & -1 & -1 \\ 1 & 0 & -2 \\ 3 & -1 & -2 \\ 2 & -1 & -1 \end{array}\right] & & B= \left[\begin{array}{cc} 3 & 3 \\ 0 & 1 \\ -2 & -2 \\ 1 & 0 \end{array}\right] & & C= \left[\begin{array}{ccc} 3 & 5 & -3 \\ 1 & 2 & -1 \end{array}\right] \end{align*}

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

Answer
\begin{equation*} BC= \left[\begin{array}{ccc} 12 & 21 & -12 \\ 1 & 2 & -1 \\ -8 & -14 & 8 \\ 3 & 5 & -3 \end{array}\right] \end{equation*}

Additional exercises available at checkit.clontz.org.