AHL 1.14 (HL)—Matrices

A matrix is a rectangular array of numbers. Its dimensions are rows × columns, and those dimensions are part of the meaning: they tell you whether an operation is defined.

Addition requires equal dimensions. For multiplication, the inner dimensions must match: an m×n matrix can multiply an n×p matrix, producing an m×p matrix. Each output entry is a row–column dot product, so matrix multiplication is generally not commutative.

For example, [[1,2],[3,4]] [5,6]^T = [17,39]^T because the first output is 1·5+2·6 and the second is 3·5+4·6. The result is 2×1, as the dimension rule predicts.

Do not multiply entries pairwise unless the question explicitly asks for a different operation. Check dimensions first, then keep the row–column order; reversing the factors may be impossible or may give a different matrix.

For A=\begin{pmatrix}a&b\c&d\end{pmatrix}, $\det A=ad-bc$ and, when this is non-zero, $A^{-1}=(ad-bc)^{-1}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$. Identity $I$ satisfies $AI=IA=A$ and zero matrix $O$ satisfies $A+O=A$. A system can be written $A\mathbf x=\mathbf b$ and solved as $\mathbf x=A^{-1}\mathbf b$ when $A$ is invertible; technology handles larger determinants/inverses, while hand work is limited to $2\times2$.