Matrices
Core concepts
Understand matrix operations, the identity and zero matrices, transposes, and determinants for (2\times2) and (3\times3) cases. Know that a square matrix is invertible exactly when its determinant is nonzero, and be able to find inverses and solve linear systems (A\mathbf{x}=\mathbf{b}) by (\mathbf{x}=A^{-1}\mathbf{b}) or by row reduction. Recognise singular matrices, conditions for unique, infinite, or no solutions, and simple applications such as transformations and networks.
Traps
Matrix multiplication is not commutative, so order matters and (AB) may exist while (BA) does not. Do not divide by matrices; multiply by the inverse on the correct side. Watch sign errors in cofactor expansion, and remember that (\det(AB)=\det A\det B) but (\det(A+B)\neq\det A+\det B). When solving systems, check whether the determinant is zero before claiming a unique solution.
Worked example
For (A=\begin{pmatrix}2&1\3&4\end{pmatrix}), (\det A=8-3=5), so (A^{-1}=\frac15\begin{pmatrix}4&-1\-3&2\end{pmatrix}). Rule of thumb: for (2\times2), swap the diagonal entries, negate the off-diagonals, and divide by the determinant.
Exam strategy
Show determinant calculations, state invertibility conditions, and verify solutions by substitution into the original system.