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Objectives

Be able to perform matrix multiplication when it is possible.
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Explore ways in which matrix multiplication acts like multiplication of numbers, and ways in which matrix multiplication is different.
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1. Is Matrix Multiplication Commutative in General?

Consider the matrices

\begin{align*} A\amp=\begin{bmatrix} 1 \amp 2 \amp 3 \end{bmatrix} \amp B\amp= \begin{bmatrix} 2 \\ -3 \\4 \end{bmatrix}\text{.} \end{align*}

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(a)

Is \(AB\) defined, and if so, what size is it?

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(b)

Is \(BA\) defined, and if so, what size is it?

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(c)

We are used to multiplication of numbers being commutative, that is, that the order in which we multiply numbers together does not matter, \(xy=yx\) for all numbers \(x\) and \(y\text{.}\) What do your answers above mean for the commutativity of matrix multiplication in general?

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2. Diagonal Matrices and the Identity Matrix.

Consider the matrices

\begin{align*} A\amp=\begin{bmatrix} 2 \amp 0 \amp 0 \\ 0 \amp -1 \amp 0 \\ 0 \amp 0 \amp 1 \end{bmatrix} \amp B\amp= \begin{bmatrix} 1 \amp 3 \amp 0 \\ 2 \amp 2 \amp -1 \\ 3 \amp 1 \amp -2 \\ \end{bmatrix}\text{.} \end{align*}

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(a)

Calculate \(AB\) and \(BA\text{.}\)

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(b)

Examine the rows of \(AB\) and the rows of \(B\text{.}\) What do you notice about the diagonal entries of \(A\) and the effect of multiplying \(B\) by \(A\) on the left?

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(c)

Examine the columns of \(BA\) and the columns of \(B\text{.}\) What do you notice about the diagonal entries of \(A\) and the effect of multiplying \(B\) by \(A\) on the right?

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(d)

The number \(1\) is called the multiplicative identity because multiplying any number \(x\) by \(1\) equals the same number you started with, \(1\cdot x = x\cdot 1 = x\text{.}\) Given an \(n\times n\) matrix \(X\text{,}\) what is the identity matrix, that is, the matrix \(I\) such that multiplying on both the left and the right leaves \(X\) unchanged, \(I\cdot X=X\cdot I = X\text{?}\)

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3. Properties of Zero and Cancellation.

Define

\begin{align*} A\amp=\begin{bmatrix} 1 \amp 2 \\ -2 \amp -4 \\ \end{bmatrix} \amp B\amp= \begin{bmatrix} 2 \amp -4 \\-1 \amp 2 \\ \end{bmatrix} \amp C\amp=\begin{bmatrix} 3 \amp 0 \\ 1 \amp 3 \end{bmatrix} \amp D\amp= \begin{bmatrix} 1 \amp 2 \\ 2 \amp 2 \end{bmatrix} \end{align*}

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(a)

With numbers, we know that if \(ab=0\text{,}\) then either \(a = 0\) or \(b=0\text{.}\)

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Compute \(AB\text{.}\) With matrices, if \(AB = 0\text{,}\) is it necessarily true that either \(A=0\) or \(B=0\text{?}\)

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(b)

When we are dealing with numbers, we know that if \(a\neq 0\) and \(ac = ad\text{,}\) then \(c=d\text{.}\)

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Compute both \(AC\) and \(AD\text{.}\) With matrices, if \(A\neq \mathbf{0}\) and \(AC = AD\text{,}\) is it necessarily true that \(C=D\text{?}\)

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4. Is Matrix Multiplication Associative? Distributive?

Consider the matrices

\begin{align*} A\amp=\begin{bmatrix} 1 \amp 2 \\ 3 \amp -2 \\ \end{bmatrix} \amp B\amp= \begin{bmatrix} 0 \amp 4 \\ 2 \amp -1 \\ \end{bmatrix} \amp C\amp= \begin{bmatrix} -1 \amp 3 \\ 4 \amp 3 \end{bmatrix}\text{.} \end{align*}

Here is a sage cell to perform (or check) the required computations.

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(a)

Compute \(AB\) and then \((AB)C\text{,}\) and record your answers.

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(b)

Compute \(BC\) and then \(A(BC)\) and record your answers.

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(c)

What does this suggest about the associativity of matrix multiplication?

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(d)