Linear Algebra · Lesson 07

Chapter practice

Bring the chapter together: set up an augmented matrix, reduce it, read the pivots and the free variables, decide how many solutions there are, and invert by row reduction. No worked example sits above the answers.

Foundations8 min readRevision 1

Read freely. Sign in when you want to save your place.

Sign in to save progress

A x = b

R_{i} \leftarrow R_{i} + c R_{j}

[A ∣ b]

rank (A)

[A ∣ I] ⟶ [I ∣ A^{- 1}]

Chapter navigation

Intuition

Bring the chapter together: set up an augmented matrix, reduce it, read the pivots and the free variables, decide how many solutions there are, and invert by row reduction. No worked example sits above the answers.

A navigator reads the same chart for the depth, the route and the hazards. An echelon form is read the same way, for consistency, freedom and rank.

Before you practise

Decide first what is asked. Solving: reduce $[A\mid\mathbf{b}]$ and substitute back. Counting solutions: compare $\operatorname{rank}A$ , $\operatorname{rank}[A\mid\mathbf{b}]$ and the number of unknowns. Describing all solutions: a particular one plus the free directions. Inverting: reduce $[A\mid I]$ .

\operatorname{rank}A=\operatorname{rank}[A\mid\mathbf{b}]\iff\text{solvable}

Reminders

Pivots are leading entries of an echelon form; their columns give the basic variables.
A row $(0\ \cdots\ 0\mid c)$ with $c\neq0$ means no solution.
Free variables: $n-\operatorname{rank}A$ when consistent.
$[A\mid I]\to[I\mid A^{-1}]$ , and a zero row on the left means no inverse.

Applications