DLA Theory

Section 9.4 Theory

In this section.

Here we will recap all of the facts we discussed in Section 9.2, as well as add in a fact from Discovery 9.1. We have already adequately discussed the ideas behind most of these facts, so for most of them we will not include a proof.

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Subsection 9.4.1 Effect of row operations on the determinant

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We begin by recording a fact that helped us in our exploration of the effect of swapping rows on the determinant.

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Lemma 9.4.1.

Any row swap can be achieved by a sequence of an odd number of adjacent row swaps.

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Proof idea.

Suppose you want to swap rows $R$ and $R'$ in a matrix using only adjacent row swaps, where $R$ appears higher in the matrix than $R'\text{,}$ and they are separated by $m$ other rows. First move $R$ down, one adjacent row swap at a time, until it is in the position just above $R'\text{.}$ Then swap $R$ and $R'\text{,}$ which are now adjacent. Finally, move $R'$ up, one adjacent row swap at a time, until it is in the original position of $R\text{.}$ Count the number of adjacent swaps that have been made as an expression in $m\text{,}$ and notice that it is odd.

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Here are all the things we learned in Discovery guide 9.1.

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Proposition 9.4.2. Determinants versus row operations.

The following are true for every square matrix.

If there is a row of zeros, then the determinant is $0\text{.}$
If two rows are swapped, then $\det(\text{new matrix}) = -\det(\text{old matrix})\text{.}$
If there are two identical rows, then the determinant is $0\text{.}$
If a row is multiplied by constant $k\text{,}$ then $\det(\text{new matrix}) = k\det(\text{old matrix})\text{.}$
If a whole matrix $A$ is scalar multiplied by a constant $k\text{,}$ then $\det (kA) = k^n\det A\text{,}$ where $n$ is the size of the matrix.
If there are two proportional rows, then the determinant is $0\text{.}$
If a multiple of one row is added to another row, then $\det(\text{new matrix}) = \det(\text{old matrix})\text{.}$

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And here are our connections between rows and columns with respect to the determinant.

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Lemma 9.4.3. Determinant of a transpose.

For every square matrix $A\text{,}$ $\det(\utrans{A}) = \det A\text{.}$

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Proposition 9.4.4. Determinants versus column operations.

The statements of Proposition 9.4.2 remain true when every instance of the word “row” is replaced by the word “column.”

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Subsection 9.4.2 Determinants of elementary matrices

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Finally, we'll record our discoveries about the determinants of elementary matrices.

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Proposition 9.4.5.

An elementary matrix corresponding to swapping rows has determinant $-1\text{.}$
An elementary matrix corresponding to multiplying a row by a constant $k$ has determinant $k\text{.}$
An elementary matrix corresponding to adding a multiple of one row to another has determinant $1\text{.}$