Matrices and Determinants
What the exam tests
Matrix questions test order, conformability, special matrices, transpose, determinant evaluation, adjoint and inverse. The exam frequently asks whether a product or inverse exists, or uses a 2×2 inverse to solve two linear equations.
Core rules
- Addition requires the same order. Multiplication (AB) requires columns of (A) equal rows of (B); generally (AB\ne BA).
- ((AB)^T=B^TA^T). A square matrix is symmetric if (A^T=A), skew-symmetric if (A^T=-A).
- For (A=\begin{pmatrix}a&b\c&d\end{pmatrix}), (|A|=ad-bc). A matrix is singular when its determinant is 0.
- (A^{-1}=\frac{1}{|A|}\operatorname{adj}A), provided (|A|\ne0). For 2×2, (\operatorname{adj}A=\begin{pmatrix}d&-b\-c&a\end{pmatrix}).
- (|AB|=|A||B|) and (|A^T|=|A|).
Worked example
For (A=\begin{pmatrix}2&1\3&4\end{pmatrix}), (|A|=8-3=5). The cofactor matrix is (\begin{pmatrix}4&-3\-1&2\end{pmatrix}), so the adjoint is its transpose, (\begin{pmatrix}4&-1\-3&2\end{pmatrix}). Hence (A^{-1}=\frac15\begin{pmatrix}4&-1\-3&2\end{pmatrix}). A quick check: (A A^{-1}=I).
Traps
(|A+B|\ne|A|+|B|) in general. Do not write the adjoint without transposing the cofactor matrix. Watch the alternating signs in cofactor expansion. State that the inverse does not exist when the determinant is zero. In matrix multiplication, do not reverse the order unless a rule permits it.