TBIL-LA Row Operations as Matrix Multiplication (MX2)

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Section 4.2 Row Operations as Matrix Multiplication (MX2)

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Learning Outcomes

Express row operations through matrix multiplication.

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Subsection 4.2.1 Class Activities

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Activity 4.2.1.

Let $A = [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] .$ Find a $3 \times 3$ matrix $B$ such that $B A = A,$ that is,

[\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}] [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}]

Check your guess using technology.

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Definition 4.2.2.

The identity matrix $I_{n}$ (or just $I$ when $n$ is obvious from context) is the $n \times n$ matrix

I_{n} = [\begin{array}{cccc} 1 & 0 & \dots & 0 \\ 0 & 1 & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & 1 \end{array}] .

It has a $1$ on each diagonal element and a $0$ in every other position.

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Fact 4.2.3.

For any square matrix $A,$ $I A = A I = A :$

[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] = [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}]

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Activity 4.2.4.

Tweaking the identity matrix slightly allows us to write row operations in terms of matrix multiplication.

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(a)

Create a matrix that doubles the third row of $A :$

[\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}] [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 2 & 2 & - 2 \end{array}]

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(b)

Create a matrix that swaps the second and third rows of $A :$

[\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}] [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 2 & 7 & - 1 \\ 1 & 1 & - 1 \\ 0 & 3 & 2 \end{array}]

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(c)

Create a matrix that adds $5$ times the third row of $A$ to the first row:

[\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}] [\begin{array}{ccc} 2 & 7 & - 1 \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}] = [\begin{array}{ccc} 2 + 5 (1) & 7 + 5 (1) & - 1 + 5 (- 1) \\ 0 & 3 & 2 \\ 1 & 1 & - 1 \end{array}]

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Fact 4.2.5.

If $R$ is the result of applying a row operation to $I,$ then $R A$ is the result of applying the same row operation to $A .$

Scaling a row: $R = [\begin{array}{ccc} c & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
Swapping rows: $R = [\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}]$
Adding a row multiple to another row: $R = [\begin{array}{ccc} 1 & 0 & c \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Such matrices can be chained together to emulate multiple row operations. In particular,

RREF (A) = R_{k} \dots R_{2} R_{1} A

for some sequence of matrices $R_{1}, R_{2}, \dots, R_{k} .$

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Activity 4.2.6.

Consider the two row operations $R_{2} \leftrightarrow R_{3}$ and $R_{1} + R_{2} \to R_{1}$ applied as follows to show $A \sim B :$

\begin{aligned} A = [\begin{array}{ccc} - 1 & 4 & 5 \\ 0 & 3 & - 1 \\ 1 & 2 & 3 \end{array}] & \sim [\begin{array}{ccc} - 1 & 4 & 5 \\ 1 & 2 & 3 \\ 0 & 3 & - 1 \end{array}] \\ \sim [\begin{array}{ccc} - 1 + 1 & 4 + 2 & 5 + 3 \\ 1 & 2 & 3 \\ 0 & 3 & - 1 \end{array}] = [\begin{array}{ccc} 0 & 6 & 8 \\ 1 & 2 & 3 \\ 0 & 3 & - 1 \end{array}] = B \end{aligned}

Express these row operations as matrix multiplication by expressing $B$ as the product of two matrices and $A :$

B = [\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}] [\begin{array}{ccc} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{array}] A

Check your work using technology.

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Subsection 4.2.2 Videos

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Figure 43. Video: Row operations as matrix multiplication

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Sample problem Example B.1.19.

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