4.2 · Matrix operations, determinants, and linear systems

Dimensions come before arithmetic

An $\displaystyle m$ by $\displaystyle n$ matrix can multiply an $\displaystyle n$ by $\displaystyle p$ matrix, producing an $\displaystyle m$ by $\displaystyle p$ result. The inner dimensions must agree; the outer dimensions give the result size.

Addition requires matching dimensions. Matrix multiplication is generally not commutative: reversing the factors can change the result or make the product undefined.

A product entry is a row-column dot product

For a matrix with rows $\displaystyle (1,2)$ and $\displaystyle (3,4)$ multiplying a column with entries $\displaystyle 5,6$, the output entries are $\displaystyle 1(5)+2(6)=17$ and $\displaystyle 3(5)+4(6)=39$.

Writing a row beside its matching column helps avoid multiplying corresponding entries without summing. Matrix-vector multiplication packages a linear system’s left sides.

The determinant detects invertibility

For a two-by-two matrix with rows $\displaystyle (a,b)$ and $\displaystyle (c,d)$, the determinant is $\displaystyle ad-bc$. A nonzero determinant gives an inverse and a unique solution for every right-hand vector.

A zero determinant means the coefficient matrix is singular. The associated system can be inconsistent or dependent, so the determinant alone does not distinguish no solution from infinitely many.

$$\det A=ad-bc$$