TBIL-LA Computing Determinants (GT2)

However, we learned in Chapter 1 that this can be tedious for large matrices. Thus, we will try to figure out how to turn the determinant of a larger matrix into the determinant of a smaller matrix.

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Activity 5.2.3.

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The following image illustrates the transformation of the unit cube by the matrix

[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 0 & 1 \end{array}] .

Figure 56. Transformation of the unit cube by the linear transformation.

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Recall that for this solid

V = B h,

where

h

is the height of the solid and

B

is the area of its parallelogram base. So what must its volume be?

$det [\begin{array}{cc} 1 & 1 \\ 1 & 3 \end{array}]$
$det [\begin{array}{cc} 1 & 0 \\ 3 & 1 \end{array}]$
$det [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}]$
$det [\begin{array}{cc} 1 & 3 \\ 0 & 0 \end{array}]$

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Fact 5.2.4.

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If row

i

contains all zeros except for a

1

on the main (upper-left to lower-right) diagonal, then both column and row

i

may be removed without changing the value of the determinant.

det [\begin{array}{cccc} 3 & 2 & - 1 & 3 \\ 0 & 1 & 0 & 0 \\ - 1 & 4 & 1 & 0 \\ 5 & 0 & 11 & 1 \end{array}] = det [\begin{array}{ccc} 3 & - 1 & 3 \\ - 1 & 1 & 0 \\ 5 & 11 & 1 \end{array}]

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Since row and column operations affect the determinants in the same way, the same technique works for a column of all zeros except for a

1

on the main diagonal.

det [\begin{array}{cccc} 3 & 0 & - 1 & 5 \\ 2 & 1 & 4 & 0 \\ - 1 & 0 & 1 & 11 \\ 3 & 0 & 0 & 1 \end{array}] = det [\begin{array}{ccc} 3 & - 1 & 5 \\ - 1 & 1 & 11 \\ 3 & 0 & 1 \end{array}]

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Put another way, if you have either a column or row from the identity matrix, you can cancel both the column and row containing the

1 .

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Warning 5.2.5.

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If the

1

is not on the main diagonal, you’ll need to use row or column swaps in order to cancel.

det [\begin{array}{cccc} 3 & 0 & - 1 & 5 \\ - 1 & 0 & 1 & 11 \\ 2 & 1 & 4 & 0 \\ 3 & 0 & 0 & 1 \end{array}] = - det [\begin{array}{cccc} 3 & 0 & - 1 & 5 \\ 2 & 1 & 4 & 0 \\ - 1 & 0 & 1 & 11 \\ 3 & 0 & 0 & 1 \end{array}] = - det [\begin{array}{ccc} 3 & - 1 & 5 \\ - 1 & 1 & 11 \\ 3 & 0 & 1 \end{array}]

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Activity 5.2.6.

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Remove an appropriate row and column of

det [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 5 & 12 \\ 3 & 2 & - 1 \end{array}]

to simplify the determinant to a

2 \times 2

determinant.

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Activity 5.2.7.

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Simplify

det [\begin{array}{ccc} 0 & 3 & - 2 \\ 2 & 5 & 12 \\ 0 & 2 & - 1 \end{array}]

to a multiple of a

2 \times 2

determinant by first doing the following:

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(a)

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Factor out a

2

from a column.

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(b)

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Swap rows or columns to put a

1

on the main diagonal.

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Activity 5.2.8.

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Simplify

det [\begin{array}{ccc} 4 & - 2 & 2 \\ 3 & 1 & 4 \\ 1 & - 1 & 3 \end{array}]

to a multiple of a

2 \times 2

determinant by first doing the following:

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(a)

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Use row/column operations to create two zeroes in the same row or column.

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(b)

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Factor/swap as needed to get a row/column of all zeroes except a

1

on the main diagonal.

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Observation 5.2.9.

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Using row/column operations, you can introduce zeros and reduce dimension to whittle down the determinant of a large matrix to a determinant of a smaller matrix.

\begin{aligned} det [\begin{array}{cccc} 4 & 3 & 0 & 1 \\ 2 & - 2 & 4 & 0 \\ - 1 & 4 & 1 & 5 \\ 2 & 8 & 0 & 3 \end{array}] & = det [\begin{array}{cccc} 4 & 3 & 0 & 1 \\ 6 & - 18 & 0 & - 20 \\ - 1 & 4 & 1 & 5 \\ 2 & 8 & 0 & 3 \end{array}] = det [\begin{array}{ccc} 4 & 3 & 1 \\ 6 & - 18 & - 20 \\ 2 & 8 & 3 \end{array}] \\ = \dots = - 2 det [\begin{array}{ccc} 1 & 3 & 4 \\ 0 & 21 & 43 \\ 0 & - 1 & - 10 \end{array}] = - 2 det [\begin{array}{cc} 21 & 43 \\ - 1 & - 10 \end{array}] \\ = \dots = - 2 det [\begin{array}{cc} - 167 & 21 \\ 0 & 1 \end{array}] = - 2 det [- 167] \\ = - 2 (- 167) det (I) = 334 \end{aligned}

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Activity 5.2.10.

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Rewrite

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

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as a multiple of a determinant of a

3 \times 3

matrix.

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Activity 5.2.11.

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Compute

det [\begin{array}{cccc} 2 & 3 & 5 & 0 \\ 0 & 3 & 2 & 0 \\ 1 & 2 & 0 & 3 \\ - 1 & - 1 & 2 & 2 \end{array}]

by using any combination of row/column operations.

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Observation 5.2.12.

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Another option is to take advantage of the fact that the determinant is linear in each row or column. This approach is called Laplace expansion or cofactor expansion.

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For example, since

[\begin{array}{ccc} 1 & 2 & 4 \end{array}] = 1 [\begin{array}{ccc} 1 & 0 & 0 \end{array}] + 2 [\begin{array}{ccc} 0 & 1 & 0 \end{array}] + 4 [\begin{array}{ccc} 0 & 0 & 1 \end{array}],

\begin{aligned} det [\begin{array}{ccc} 2 & 3 & 5 \\ - 1 & 3 & 5 \\ 1 & 2 & 4 \end{array}] & = 1 det [\begin{array}{ccc} 2 & 3 & 5 \\ - 1 & 3 & 5 \\ 1 & 0 & 0 \end{array}] + 2 det [\begin{array}{ccc} 2 & 3 & 5 \\ - 1 & 3 & 5 \\ 0 & 1 & 0 \end{array}] + 4 det [\begin{array}{ccc} 2 & 3 & 5 \\ - 1 & 3 & 5 \\ 0 & 0 & 1 \end{array}] \\ = - 1 det [\begin{array}{ccc} 5 & 3 & 2 \\ 5 & 3 & - 1 \\ 0 & 0 & 1 \end{array}] - 2 det [\begin{array}{ccc} 2 & 5 & 3 \\ - 1 & 5 & 3 \\ 0 & 0 & 1 \end{array}] + 4 det [\begin{array}{ccc} 2 & 3 & 5 \\ - 1 & 3 & 5 \\ 0 & 0 & 1 \end{array}] \\ = - det [\begin{array}{cc} 5 & 3 \\ 5 & 3 \end{array}] - 2 det [\begin{array}{cc} 2 & 5 \\ - 1 & 5 \end{array}] + 4 det [\begin{array}{cc} 2 & 3 \\ - 1 & 3 \end{array}] \end{aligned}

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Observation 5.2.13.

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Recall the formula for a

2 \times 2

determinant found in Observation 5.1.28:

det [\begin{array}{cc} a & b \\ c & d \end{array}] = a d - b c .

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There are formulas and algorithms for the determinants of larger matrices, but they can be pretty tedious to use. For example, writing out a formula for a

4 \times 4

determinant would require 24 different terms!

det [\begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array}] = a_{11} (a_{22} (a_{33} a_{44} - a_{43} a_{34}) - a_{23} (a_{32} a_{44} - a_{42} a_{34}) + \dots) + \dots

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Activity 5.2.14.

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Based on the previous activities, which technique is easier for computing determinants?

Memorizing formulas.
Using row/column operations.
Laplace expansion.
Some other technique.

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Activity 5.2.15.

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Use your preferred technique to compute

det [\begin{array}{cccc} 4 & - 3 & 0 & 0 \\ 1 & - 3 & 2 & - 1 \\ 3 & 2 & 0 & 3 \\ 0 & - 3 & 2 & - 2 \end{array}] .

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Insight 5.2.16.

You can check your answers using technology.

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det([4,-3,0,0; 1,-3,2,-1; 3,2,0,3; 0,-3,2,-2])

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Subsection 5.2.3 Individual Practice

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Activity 5.2.17.

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A diagonal matrix is a matrix that has zeroes in every position except (possibly) the main upper-left to lower-right diagonal. A matrix is upper (resp. lower) triangular if every entry below (resp. above) the main diagonal is zero.

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