Elementary row operations

MATElementary Operations & the Inverse

Elementary row operations are the three reversible moves on a matrix's rows used to simplify it to echelon or identity form. They underpin both finding inverses and solving linear systems, and ISC examines them as a 'list each operation' exercise demanding clear notation.

Mastery means reaching the target form efficiently while writing every step in standard symbols.

The three elementary row operations

R_i \leftrightarrow R_j, \quad R_i \to kR_i\ (k \ne 0), \quad R_i \to R_i + kR_j

Row interchange; multiply a row by a nonzero scalar

k

; add

k

times row

j

to row

i

. These are the only permitted operations.

Row-echelon (upper-triangular) target

\begin{pmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{pmatrix}

Leading entries (pivots) step down to the right; all entries below each pivot are 0.

*

denotes any value.

Reduction toward identity

A \xrightarrow{\text{row ops}} I

When

A

is non-singular, a finite sequence of elementary row operations transforms it to the identity

I

Only three operations are allowed: swap, scale by nonzero $k$ , and add a multiple of another row.
Each elementary row operation is reversible, so the row space and solution set are preserved.
Aim for pivots: make a leading 1 in a column, then eliminate the entries below (echelon) or below and above (reduced).
Work top-to-bottom for echelon form, clearing column 1 first, then column 2, and so on.
Write every operation in standard notation, e.g. $R_2 \to R_2 - 2R_1$ , so the marker can follow each step.
Scaling must use a NONZERO multiplier; multiplying a row by 0 is not an elementary operation.
A row of all zeros appearing means the rows are linearly dependent (matrix is singular).

Where the marks go

Multiplying a row by 0 or by a non-scalar, which is not a valid elementary operation.
Combining two operations into one step without recording them, losing method marks.
Changing the wrong row in $R_i \to R_i + kR_j$ (altering $R_j$ instead of $R_i$ ).
Not writing the operation symbol for each step, so the working cannot be verified.

How the board asks it

Numericalthe three elementary row operations
Reduce the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ 3 & 7 & 9 \end{pmatrix}$ to row-echelon form using elementary row operations, writing each operation in standard notation such as $R_2 \to R_2 - 2R_1$ .
Numericalreduction toward the identity matrix
Using elementary row operations, find the inverse of the matrix $A = \begin{pmatrix} 2 & 1 \\ 7 & 4 \end{pmatrix}$ .
Numericalechelon form preserves the solution set
Solve the system $x + 2y + z = 4$ , $2x + 3y - z = 1$ , $3x - y + 2z = 5$ by reducing the augmented matrix to echelon form using elementary row operations.
Give reasonsa zero row signals linear dependence
While reducing a $3 \times 3$ matrix to echelon form a student obtains a complete row of zeros. State what this implies about the matrix and give a reason.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.