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.

Subsection 9.4.1 Effect of row operations on the determinant

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

Lemma 9.4.1.

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

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.

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

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
\begin{equation*} \det(\text{new matrix}) = -\det(\text{old matrix})\text{.} \end{equation*}
If there are two identical rows, then the determinant is \(0\text{.}\)
If a row is multiplied by constant \(k\text{,}\) then
\begin{equation*} \det(\text{new matrix}) = k\det(\text{old matrix})\text{.} \end{equation*}
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
\begin{equation*} \det(\text{new matrix}) = \det(\text{old matrix})\text{.} \end{equation*}

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

Lemma 9.4.3. Determinant of a transpose.

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

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.”

Subsection 9.4.2 Determinants of elementary matrices

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

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 \(1\text{.}\)

Prev Top Next