Notation for Row Operations

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We summarize the notation to keep track of the precise row operations being used.

In the process of putting the ACM into reduced row echelon form, it is often desirable to keep track of the precise row operations being used. For this, some notation is useful. The following summarizes the notation we use in this course.

Type I operation

What it does: switches $i^{th}$ and $j^{th}$ rows.

Indicated by: $R_i\leftrightarrow R_j$

Type II operation

What it does: multiplies $i^{th}$ row by $r\ne 0$ .

Indicated by: $r\cdot R_i$

Type III operation

What it does: adds $a$ times the $i^{th}$ row to the $j^{th}$ row

Indicated by: $R_j := R_j + a\cdot R_i$ or “ $a\cdot R_i$ added to $R_j$ ”

An example of a specific type I operation switching the 2nd and 3rd rows might be $\begin {bmatrix} 3 & 2 & 7\\ 4 & 1 & 9\\ 1 & 0 & -2 \end {bmatrix} \xrightarrow {R_2\leftrightarrow R_3} \begin {bmatrix} 3 & 2 & 7\\ 1 & 0 & -2\\ 4 & 1 & 9 \end {bmatrix}$

Suppose we started with the same matrix, but performed a type II operation instead, where the second row is multiplied by $3$ . Then using the above conventions, we would denote that by $\begin {bmatrix} 3 & 2 & 7\\ 4 & 1 & 9\\ 1 & 0 & -2 \end {bmatrix} \xrightarrow {3\cdot R_2} \begin {bmatrix} 3 & 2 & 7\\ 12 & 3 & 27\\ 1 & 0 & -2 \end {bmatrix}$

Suppose we started with the same matrix, but performed a type III operation, where the second row is multiplied by $-4$ and added to the first row. Then using the above conventions, we would denote that by $\begin {bmatrix} 3 & 2 & 7\\ 4 & 1 & 9\\ 1 & 0 & -2 \end {bmatrix} \xrightarrow {R_1 := R_1 + (-4)\cdot R_2} \begin {bmatrix} -13 & -2 & -29\\ 4& 1 & 9\\ 1 & 0 & -2 \end {bmatrix}$

In the following exercises, fill in the blank with the indicated row operation needed to produce the matrix on the right from the matrix on the left.

$\begin {bmatrix} -1 & 3 & 2\\ 9 & 7 & 6\\ -3 & 5 & -7 \end {bmatrix} \xrightarrow {\phantom {xxxxxxxx}} \begin {bmatrix} -3 & 5 & -7\\ 9 & 7 & 6\\ -1 & 3 & 2 \end {bmatrix}$

$\begin {bmatrix} -1 & 3 & 2\\ 9 & 7 & 6\\ -3 & 5 & -7 \end {bmatrix} \xrightarrow {\phantom {xxxxxxxx}} \begin {bmatrix} -1 & 3 & 2\\ 9 & 7 & 6\\ 6 & -10 & 14 \end {bmatrix}$

$\begin {bmatrix} -1 & 3 & 2\\ 9 & 7 & 6\\ -3 & 5 & -7 \end {bmatrix} \xrightarrow {\phantom {xxxxxxxx}} \begin {bmatrix} -1 & 3 & 2\\ 9 & 7 & 6\\ -2 & 2 & -9 \end {bmatrix}$

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
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*	$\cdot$
infinity	$\infty$
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Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Type I operation

Type II operation

Type III operation

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