Matrix algebra and transformations

What you’ll learn

• Multiply and invert two-by-two matrices.
• Interpret transformations and their composition.

Understand the idea

Matrix multiplication combines rows of the left matrix with columns of the right. The determinant of a two-by-two matrix decides whether it has an inverse; a zero determinant means the associated plane transformation collapses area and cannot be reversed uniquely.

The columns of a transformation matrix are the images of $(1,0)$ and $(0,1)$. For column vectors, $AB$ applies $B$ first and then $A$. Rotations, reflections, stretches and enlargements can be represented this way, and the absolute determinant is the area scale factor.

Key formulae

$$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$$

$$A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$

$$R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$$

$$\mathbf{x}'=AB\mathbf{x}$$

Worked example

Find the inverse of $A=\begin{pmatrix}2&1\\1&3\end{pmatrix}$ and use it to solve $A\binom{x}{y}=\binom{7}{11}$.

1. The determinant is $6-1=5$.
2. Thus $A^{-1}=\frac15\begin{pmatrix}3&-1\\-1&2\end{pmatrix}$.
3. Multiply by the right-hand column to get $\frac15\binom{21-11}{-7+22}=\binom23$.

Answer: $x=2$, $y=3$.