Matrix Algebra Topic Review summary

Key learning points

• Matrix multiplication is not commutative; check conformability.
• \\det\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}=ad-bc; singular if determinant 0.
• Inverse \\frac{1}{ad-bc}\\begin{pmatrix}d&-b\\\\-c&a\\end{pmatrix} when $\\det\\neq0$.
• Transformations: matrices for reflections, rotations, enlargements, shears.
• Invariant points/lines solve $(\\mathbf{M}-I)\\mathbf{x}=\\mathbf{0}$ style conditions.

Worked example 1 — Det

\\det\\begin{pmatrix}2&1\\\\4&3\\end{pmatrix}.

Working

$ad-bc$.

Worked example 2 — Inverse

Inverse of \\begin{pmatrix}2&1\\\\4&3\\end{pmatrix}.

Working

Use formula with $\\det=2$.

Worked example 3 — Practice point 1

Apply this idea from the Topic Review: Matrix multiplication is not commutative; check conformability.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

\\\\det\\\\begin{pmatrix}a&b\\\\\\\\c&d\\\\end{pmatrix}=ad-bc; singular if determinant 0.

Working

Attempt without notes, then check the answers PDF.

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.