TBIL-LA Row Operations and Determinants (GT1)

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Section 5.1 Row Operations and Determinants (GT1)

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Learning Outcomes

Describe how a row operation affects the determinant of a matrix.

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Subsection 5.1.1 Class Activities

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Activity 5.1.1.

The image in Figure 46 illustrates how the linear transformation $T : R^{2} \to R^{2}$ given by the standard matrix $A = [\begin{array}{cc} 2 & 0 \\ 0 & 3 \end{array}]$ transforms the unit square.

Figure 46. Transformation of the unit square by the matrix $A .$

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(a)

What are the lengths of $A {\vec{e}}_{1}$ and $A {\vec{e}}_{2} ?$

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(b)

What is the area of the transformed unit square?

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Activity 5.1.2.

The image below illustrates how the linear transformation $S : R^{2} \to R^{2}$ given by the standard matrix $B = [\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}]$ transforms the unit square.

Figure 47. Transformation of the unit square by the matrix $B$

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(a)

What are the lengths of $B {\vec{e}}_{1}$ and $B {\vec{e}}_{2} ?$

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(b)

What is the area of the transformed unit square?

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Observation 5.1.3.

It is possible to find two nonparallel vectors that are scaled but not rotated by the linear map given by $B .$

B {\vec{e}}_{1} = [\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}] [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 2 \\ 0 \end{matrix}] = 2 {\vec{e}}_{1}

B [\begin{matrix} \frac{3}{4} \\ \frac{1}{2} \end{matrix}] = [\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}] [\begin{matrix} \frac{3}{4} \\ \frac{1}{2} \end{matrix}] = [\begin{matrix} 3 \\ 2 \end{matrix}] = 4 [\begin{matrix} \frac{3}{4} \\ \frac{1}{2} \end{matrix}]

Figure 48. Certain vectors are stretched out without being rotated.

The process for finding such vectors will be covered later in this chapter.

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Observation 5.1.4.

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.

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 5.1.5.

We will define the determinant of a square matrix $B,$ or $det (B)$ 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 5.1.6.

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.

The value for $det ([{\vec{e}}_{1} {\vec{e}}_{2}]) = det (I)$ is:

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Activity 5.1.7.

The transformation of the unit square by the standard matrix $[\vec{v} \vec{v}]$ is illustrated below: both $T ({\vec{e}}_{1}) = T ({\vec{e}}_{2}) = \vec{v} .$ If $det ([\vec{v} \vec{v}])$ 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.

The value of $det ([\vec{v} \vec{v}])$ is:

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Activity 5.1.8.

The transformations of the unit square by the standard matrices $[\vec{v} \vec{w}]$ and $[c \vec{v} \vec{w}]$ are illustrated below. Describe the value of $det ([c \vec{v} \vec{w}]) .$

Figure 53. The parallelograms generated by $\vec{v}$ and $\vec{w}$ / $c \vec{w}$

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.

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Consider the vectors

\vec{u},

\vec{v},

\vec{u} + \vec{v},

and

\vec{w}

displayed below. Each pair of vectors generates a parallelogram, and the area of each parallelogram can be described in terms of determinants.

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Figure 54. The vectors $\vec{u},$ $\vec{v},$ $\vec{u} + \vec{v}$ and $\vec{w}$

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For example,

det ([\vec{u} \vec{w}])

represents the shaded area shown below.

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Figure 55. Parallelogram generated by $\vec{u}$ and $\vec{w}$

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Similarly,

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

represents the shaded area shown below.

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Figure 56. Parallelogram generated by $\vec{v}$ and $\vec{w}$

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Activity 5.1.9.

The paralellograms generated by the standard matrices $[\vec{u} \vec{w}],$ $[\vec{v} \vec{w}]$ and $[\vec{u} + \vec{v} \vec{w}]$ are illustrated below.

Figure 57. Parallelogram generated by $\vec{u} + \vec{v}$ and $\vec{w}$

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 5.1.10.

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.

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

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Observation 5.1.11.

The determinant must also satisfy other properties. Consider $det ([\vec{v} \vec{w} + c \vec{v}])$ and $det ([\vec{v} \vec{w}]) .$

Figure 58. Parallelogram built by $\vec{w} + c \vec{v}$ and $\vec{w}$

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 5.1.12.

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 59. Reflection of a parallelogram as a result of swapping columns.

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Observation 5.1.13.

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 5.1.14.

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 5.1.15.

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 60. Area changing under the composition of two linear maps

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

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Fact 5.1.16.

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 5.1.17.

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 5.1.18.

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 5.1.19.

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)

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)

Find $det R$ by comparing with the previous slide.

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(c)

If $C \in M_{4, 4}$ is a matrix with $det (C) = - 3,$ find

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

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Activity 5.1.20.

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)

Find a matrix $R$ such that $B = R A,$ by applying the same row operation to $I .$

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(b)

If $C \in M_{4, 4}$ is a matrix with $det (C) = 5,$ find $det (R C) .$

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Activity 5.1.21.

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)

Find a matrix $R$ such that $B = R A .$

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(b)

If $C \in M_{4, 4}$ is a matrix with $det (C) = - 7,$ find $det (R C) .$

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Remark 5.1.22.

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 5.1.23.