TBIL-LA Row Operations and Determinants (GT1)

Notice that while a linear map can transform vectors in various ways, linear maps always transform parallelograms into parallelograms, and these areas are always transformed by the same factor: in the case of

B = [\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}],

this factor is

8 .

Figure 49. A linear map transforming parallelograms into parallelograms.

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Since this change in area is always the same for a given linear map, it will be equal to the value of the transformed unit square (which begins with area

1

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Remark 5.1.6.

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We will define the determinant of a square matrix

B,

det (B)

for short, to be the factor by which

B

scales areas. In order to figure out how to compute it, we first figure out the properties it must satisfy.

Figure 50. The linear transformation $B$ scaling areas by a constant factor, which we call the determinant

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Activity 5.1.7.

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The transformation of the unit square by the standard matrix

[{\vec{e}}_{1} {\vec{e}}_{2}] = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = I

is illustrated below. If

det ([{\vec{e}}_{1} {\vec{e}}_{2}]) = det (I)

is the area of resulting parallelogram, what is the value of

det ([{\vec{e}}_{1} {\vec{e}}_{2}]) = det (I) ?

Figure 51. The transformation of the unit square by the identity matrix.

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The value for

det ([{\vec{e}}_{1} {\vec{e}}_{2}]) = det (I)

is:

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Activity 5.1.8.

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The transformation of the unit square by the standard matrix

[\vec{v} \vec{v}]

is illustrated below: both

T ({\vec{e}}_{1}) = T ({\vec{e}}_{2}) = \vec{v} .

det ([\vec{v} \vec{v}])

is the area of the generated parallelogram, what is the value of

det ([\vec{v} \vec{v}]) ?

Figure 52. Transformation of the unit square by a matrix with identical columns.

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The value of

det ([\vec{v} \vec{v}])

is:

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Activity 5.1.9.

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Describe the value of

det ([c \vec{v} \vec{w}]) :

$det ([\vec{v} \vec{w}])$
$c det ([\vec{v} \vec{w}])$
$c^{2} det ([\vec{v} \vec{w}])$
Cannot be determined from this information.

Answer.

c det ([\vec{v} \vec{w}]) :

the value of

c

directly scales the determinant area because it scales the base of the parallelogram.

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Activity 5.1.10.

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Describe the value of

det ([\vec{u} + \vec{v} \vec{w}]) .

$det ([\vec{u} \vec{w}]) = det ([\vec{v} \vec{w}])$
$det ([\vec{u} \vec{w}]) + det ([\vec{v} \vec{w}])$
$det ([\vec{u} \vec{w}]) det ([\vec{v} \vec{w}])$
Cannot be determined from this information.

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Definition 5.1.11.

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The determinant is the unique function

det : M_{n, n} \to R

satisfying these properties:

$det (I) = 1$
$det (A) = 0$ whenever two columns of the matrix are identical.
$det [\dots c \vec{v} \dots] = c det [\dots \vec{v} \dots],$ assuming no other columns change.
$det [\dots \vec{v} + \vec{w} \dots] = det [\dots \vec{v} \dots] + det [\dots \vec{w} \dots],$ assuming no other columns change.

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Note that these last two properties together can be phrased as “The determinant is linear in each column.”

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Observation 5.1.12.

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The determinant must also satisfy other properties. Consider

det ([\vec{v} \vec{w} + c \vec{v}])

and

det ([\vec{v} \vec{w}]) .

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The base of both parallelograms is

\vec{v},

while the height has not changed, so the determinant does not change either. This can also be proven using the other properties of the determinant:

\begin{aligned} det ([\vec{v} + c \vec{w} \vec{w}]) & = det ([\vec{v} \vec{w}]) + det ([c \vec{w} \vec{w}]) \\ = det ([\vec{v} \vec{w}]) + c det ([\vec{w} \vec{w}]) \\ = det ([\vec{v} \vec{w}]) + c \cdot 0 \\ = det ([\vec{v} \vec{w}]) \end{aligned}

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Remark 5.1.13.

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Swapping columns may be thought of as a reflection, which is represented by a negative determinant. For example, the following matrices transform the unit square into the same parallelogram, but the second matrix reflects its orientation.

A = [\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}] det A = 8 B = [\begin{array}{cc} 3 & 2 \\ 4 & 0 \end{array}] det B = - 8

Figure 53. Reflection of a parallelogram as a result of swapping columns.

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Observation 5.1.14.

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The fact that swapping columns multiplies determinants by a negative may be verified by adding and subtracting columns.

\begin{aligned} det ([\vec{v} \vec{w}]) & = det ([\vec{v} + \vec{w} \vec{w}]) \\ = det ([\vec{v} + \vec{w} \vec{w} - (\vec{v} + \vec{w})]) \\ = det ([\vec{v} + \vec{w} - \vec{v}]) \\ = det ([\vec{v} + \vec{w} - \vec{v} - \vec{v}]) \\ = det ([\vec{w} - \vec{v}]) \\ = - det ([\vec{w} \vec{v}]) \end{aligned}

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Fact 5.1.15.

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To summarize, we’ve shown that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant in the following way:

Multiplying a column by a scalar multiplies the determinant by that scalar:

$c det ([\dots \vec{v} \dots]) = det ([\dots c \vec{v} \dots])$
Swapping two columns changes the sign of the determinant:

$det ([\dots \vec{v} \dots \vec{w} \dots]) = - det ([\dots \vec{w} \dots \vec{v} \dots])$
Adding a multiple of a column to another column does not change the determinant:

$det ([\dots \vec{v} \dots \vec{w} \dots]) = det ([\dots \vec{v} + c \vec{w} \dots \vec{w} \dots])$

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Activity 5.1.16.

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The transformation given by the standard matrix

A

scales areas by

4,

and the transformation given by the standard matrix

B

scales areas by

3 .

By what factor does the transformation given by the standard matrix

A B

scale areas?

Figure 54. Area changing under the composition of two linear maps

$1$
$7$
$12$
Cannot be determined

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Fact 5.1.17.

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Since the transformation given by the standard matrix

A B

is obtained by applying the transformations given by

A

and

B,

it follows that

det (A B) = det (A) det (B) = det (B) det (A) = det (B A) .

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Remark 5.1.18.

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Recall that row operations may be produced by matrix multiplication.

Multiply the first row of $A$ by $c :$ $[\begin{array}{cccc} c & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] A$
Swap the first and second row of $A :$ $[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] A$
Add $c$ times the third row to the first row of $A :$ $[\begin{array}{cccc} 1 & 0 & c & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] A$

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Fact 5.1.19.

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The determinants of row operation matrices may be computed by manipulating columns to reduce each matrix to the identity:

Scaling a row: $det [\begin{array}{cccc} c & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] = c det [\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] = c$
Swapping rows: $det [\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] = - 1 det [\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] = - 1$
Adding a row multiple to another row: $det [\begin{array}{cccc} 1 & 0 & c & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] = det [\begin{array}{cccc} 1 & 0 & c - 1 c & 0 \\ 0 & 1 & 0 - 0 c & 0 \\ 0 & 0 & 1 - 0 c & 0 \\ 0 & 0 & 0 - 0 c & 1 \end{array}] = det (I) = 1$

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Activity 5.1.20.

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Consider the row operation

R_{1} + 4 R_{3} \to R_{1}

applied as follows to show

A \sim B :

A = [\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}] \sim [\begin{array}{cccc} 1 + 4 (9) & 2 + 4 (10) & 3 + 4 (11) & 4 + 4 (12) \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}] = B

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(a)

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Find a matrix

R

such that

B = R A,

by applying the same row operation to

I = [\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] .

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(b)

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Find

det R

by comparing with the previous slide.

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(c)

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C \in M_{4, 4}

is a matrix with

det (C) = - 3,

find

det (R C) = det (R) det (C) .

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Activity 5.1.21.

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Consider the row operation

R_{1} \leftrightarrow R_{3}

applied as follows to show

A \sim B :

A = [\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}] \sim [\begin{array}{cccc} 9 & 10 & 11 & 12 \\ 5 & 6 & 7 & 8 \\ 1 & 2 & 3 & 4 \\ 13 & 14 & 15 & 16 \end{array}] = B

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(a)

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Find a matrix

R

such that

B = R A,

by applying the same row operation to

I .

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(b)

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C \in M_{4, 4}

is a matrix with

det (C) = 5,

find

det (R C) .

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Activity 5.1.22.

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Consider the row operation

3 R_{2} \to R_{2}

applied as follows to show

A \sim B :

A = [\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}] \sim [\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 3 (5) & 3 (6) & 3 (7) & 3 (8) \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}] = B

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(a)

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Find a matrix

R

such that

B = R A .

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(b)

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C \in M_{4, 4}

is a matrix with

det (C) = - 7,

find

det (R C) .

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Activity 5.1.23.

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Let

A

be any

4 \times 4

matrix with determinant

2 .

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(a)

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Let

B

be the matrix obtained from

A

by applying the row operation

R_{1} - 5 R_{3} \to R_{1} .

What is

\det B ?

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-4
-2
2
10

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(b)

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Let

M

be the matrix obtained from

A

by applying the row operation

R_{3} \leftrightarrow R_{1} .

What is

\det M ?

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-4
-2
2
10

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(c)

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Let

P

be the matrix obtained from

A

by applying the row operation

2 R_{4} \to R_{4} .

What is

\det P ?

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-4
-2
2
10

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Remark 5.1.24.

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Recall that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant:

Multiplying columns by scalars:

$det ([\dots c \vec{v} \dots]) = c det ([\dots \vec{v} \dots])$
Swapping two columns:

$det ([\dots \vec{v} \dots \vec{w} \dots]) = - det ([\dots \vec{w} \dots \vec{v} \dots])$
Adding a multiple of a column to another column:

$det ([\dots \vec{v} \dots \vec{w} \dots]) = det ([\dots \vec{v} + c \vec{w} \dots \vec{w} \dots])$

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Remark 5.1.25.

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The determinants of row operation matrices may be computed by manipulating columns to reduce each matrix to the identity:

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

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Fact 5.1.26.

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Thus we can also use both row operations to simplify determinants:

Multiplying rows by scalars:

$det [\begin{matrix} ⋮ \\ c R \\ ⋮ \end{matrix}] = c det [\begin{matrix} ⋮ \\ R \\ ⋮ \end{matrix}]$
Swapping two rows:

$det [\begin{matrix} ⋮ \\ R \\ ⋮ \\ S \\ ⋮ \end{matrix}] = - det [\begin{matrix} ⋮ \\ S \\ ⋮ \\ R \\ ⋮ \end{matrix}]$
Adding multiples of rows/columns to other rows:

$det [\begin{matrix} ⋮ \\ R \\ ⋮ \\ S \\ ⋮ \end{matrix}] = det [\begin{matrix} ⋮ \\ R + c S \\ ⋮ \\ S \\ ⋮ \end{matrix}]$

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Activity 5.1.27.

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Complete the following derivation for a formula calculating

2 \times 2

determinants:

\begin{aligned} det [\begin{array}{cc} a & b \\ c & d \end{array}] & = ? det [\begin{array}{cc} 1 & b / a \\ c & d \end{array}] \\ = ? det [\begin{array}{cc} 1 & b / a \\ c - c & d - b c / a \end{array}] \\ = ? det [\begin{array}{cc} 1 & b / a \\ 0 & d - b c / a \end{array}] \\ = ? det [\begin{array}{cc} 1 & b / a \\ 0 & 1 \end{array}] \\ = ? det [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] \\ = ? det I \\ = ? \end{aligned}

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Observation 5.1.28.

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So we may compute the determinant of

[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}]

by using determinant properties to manipulate its rows/columns to reduce the matrix to

I :

\begin{aligned} det [\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}] & = 2 det [\begin{array}{cc} 1 & 2 \\ 2 & 3 \end{array}] \\ = 2 det [\begin{array}{cc} 1 & 2 \\ 0 & - 1 \end{array}] \\ = - 2 det [\begin{array}{cc} 1 & - 2 \\ 0 & 1 \end{array}] \\ = - 2 det [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] \\ = - 2 \end{aligned}

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Or we may use a formula:

det [\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}] = (2) (3) - (4) (2) = - 2

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Subsection 5.1.3 Individual Practice

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Activity 5.1.29.

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Suppose we have a linear transformation

T : R^{2} \to R^{2} .

Given some shape

S

in the plane

R^{2},

we can use to

T

to transform it into some new shape

T (S) .

Consider the following questions about properties that may or may not be preserved by

T .

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(a)

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S

is a straight line segment, explain why

T (S)

is also a straight line segment.

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(b)

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S

is a straight line segment, does

T (S)

necessarily have to have the same length as that of

S ?

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(c)

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S

is a triangle, explain why

T (S)

is also a triangle.

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(d)

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Continuing as above, do the angles of

T (S)

necessarily have to be the same as those of

S ?

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Subsection 5.1.4 Videos

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Figure 55. Video: Row operations, matrix multiplication, and determinants

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Exercises 5.1.5 Exercises

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Exercises available at https://tbil.org/linear-algebra/preview/exercises/#/bank/GT1/.

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Subsection 5.1.6 Mathematical Writing Explorations

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Exploration 5.1.30.

Prove or disprove. The determinant is a linear operator on the vector space of $n \times n$ matrices.
Find a matrix that will double the area of a region in $R^{2} .$
Find a matrix that will triple the area of a region in $R^{2} .$
Find a matrix that will halve the area of a region in $R^{2} .$

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Subsection 5.1.7 Sample Problem and Solution

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Sample problem Example B.1.22.

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