Matrix Algebra summary

Key learning points

• Matrix multiplication is associative but not commutative in general.
• $2\times2$ inverse exists when $ad-bc\ne0$.
• Determinant zero ⇒ singular (no inverse).
• Transformations: multiply position vectors on the left.

Worked example 1 — Determinant

Find $\det\begin{pmatrix}2&1\\0&3\end{pmatrix}$.

Working

$ad-bc=6$.

Worked example 2 — Matrix multiplication is associative but not commutativ

Use this method from the Matrix Algebra sheet: Matrix multiplication is associative but not commutative in general.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — inverse exists when

Use this method from the Matrix Algebra sheet: $2\times2$ inverse exists when $ad-bc\ne0$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Determinant zero ⇒ singular (no inverse)

Use this method from the Matrix Algebra sheet: Determinant zero ⇒ singular (no inverse).

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Transformations: multiply position vectors on the left

Use this method from the Matrix Algebra sheet: Transformations: multiply position vectors on the left.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.