TBIL-LA Row Operations as Matrix Multiplication (MX4)

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Section 4.4 Row Operations as Matrix Multiplication (MX4)

Learning Outcomes

Express row operations through matrix multiplication.

Subsection 4.4.1 Warm Up

Activity 4.4.1.

Given a linear transformation

T,

how did we define its standard matrix

A ?

How do we compute the standard matrix

A

from

T ?

Subsection 4.4.2 Class Activities

Activity 4.4.2.

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

(a)

Which of these tweaks of the identity matrix yields a matrix that doubles the third row of

A

when left-multiplying? (

2 R_{3} \to R_{3}

)

[\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}]

$[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
$[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}]$
$[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}]$
$[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}]$

(b)

Which of these tweaks of the identity matrix yields a matrix that swaps the first and third rows of

A

when left-multiplying? (

R_{1} \leftrightarrow R_{3}

)

[\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}]

$[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}]$
$[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}]$
$[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}]$
$[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}]$

(c)

Which of these tweaks of the identity matrix yields a matrix that adds

5

times the third row of

A

to the first row when left-multiplying? (

R_{1} + 5 R_{3} \to R_{1}

)

[\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}]

$[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}]$
$[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
$[\begin{array}{ccc} 5 & 5 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$
$[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}]$

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Fact 4.4.3.

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} .

Activity 4.4.4.

What would happen if you right-multiplied by the tweaked identity matrix rather than left-multiplied?

The manipulated rows would be reversed.
Columns would be manipulated instead of rows.
The entries of the resulting matrix would be rotated 180 degrees.

Activity 4.4.5.

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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Activity 4.4.6.

Let

A

be any

4 \times 4

matrix.

(a)

Give a

4 \times 4

matrix

M

that may be used to perform the row operation

- 5 R_{2} \to R_{2} .

(b)

Give a

4 \times 4

matrix

Y

that may be used to perform the row operation

R_{2} \leftrightarrow R_{3} .

(c)

Use matrix multiplication to describe the matrix obtained by applying

- 5 R_{2} \to R_{2}

and then

R_{2} \leftrightarrow R_{3}

to

A

(note the order).

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Subsection 4.4.3 Individual Practice

Activity 4.4.7.

Consider the matrix

A = [\begin{matrix} 2 & 6 & - 1 & 6 \\ 1 & 3 & - 1 & 2 \\ - 1 & - 3 & 2 & 0 \end{matrix}] .

Illustrate Fact 4.4.3 by finding row operation matrices

R_{1}, \dots, R_{k}

for which

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

If you and a teammate were to do this independently, would you necessarily come up with the same sequence of matrices

R_{1}, \dots, R_{k} ?

Subsection 4.4.4 Videos

Figure 45. Video: Row operations as matrix multiplication

Exercises 4.4.5 Exercises

Exercises available at https://tbil.org/linear-algebra/preview/exercises/#/bank/MX4/.

Subsection 4.4.6 Sample Problem and Solution

Sample problem Example B.1.21.