Matrices & linear transformations

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• Matrices encode linear rules. Add or subtract only matrices of the same order; multiply $AB$ only when the number of columns of $A$ matches the number of rows of $B$. In general $AB\ne BA$, so order is part of the method (CP-3.1).
• The zero matrix plays the role of $0$ and the identity matrix $I$ plays the role of $1$: $AI=IA=A$. A transformation followed by its inverse has matrix $A^{-1}A=I$ (CP-3.2 and CP-3.6).
• For a 2-D transformation, the columns of the matrix are the images of $\binom10$ and $\binom01$. If transformation $A$ happens first and $B$ second, the combined matrix is $BA$, because $\mathbf{x}\mapsto A\mathbf{x}\mapsto BA\mathbf{x}$ (CP-3.3).
• For a 3-D transformation, the three columns are the images of the standard basis vectors $\begin{pmatrix}1\\0\\0\end{pmatrix}$, $\begin{pmatrix}0\\1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\0\\1\end{pmatrix}$ (CP-3.3).
• Worked example (CP-3.3): reflection in the plane $z=0$ maps $(x,y,z)$ to $(x,y,-z)$, so its matrix is $R=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}$. For example, $R\begin{pmatrix}2\\-3\\4\end{pmatrix}=\begin{pmatrix}2\\-3\\-4\end{pmatrix}$, while $R^2=I$ confirms that reflecting twice restores every point.
• Invariant points satisfy $A\mathbf{x}=\mathbf{x}$, so solve $(A-I)\mathbf{x}=\mathbf0$. For an invariant line through the origin, transform $(x,mx)$ and require its image to remain on the line. For a general line $y=mx+c$, transform $(x,mx+c)$ and impose $Y=mX+c$ for every $x$; if every point is fixed as well as remaining on the line, it is a line of invariant points (CP-3.4).
• The determinant is a signed scale factor. In 2-D, areas are multiplied by $|\det A|$; in 3-D, volumes are multiplied by $|\det A|$. A negative determinant reverses orientation (CP-3.5).
• A matrix is singular exactly when its determinant is zero. Then no inverse exists and dimensions have been collapsed. For $A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$, $\det A=1$ and $A^{-1}=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}$ (CP-3.6).
• To solve $A\mathbf{x}=\mathbf{b}$ with a non-singular $3\times3$ matrix, use $\mathbf{x}=A^{-1}\mathbf{b}$. Keep the exact matrix visible: method marks depend on using the inverse, not merely reporting calculator output (CP-3.7).
• Three simultaneous linear equations represent three planes. A unique solution is one common point; infinitely many solutions give a common line or coincident plane; no solution means the planes have no common point (CP-3.8).
• Worked example: the reflection in $y=x$ has matrix $\begin{pmatrix}0&1\\1&0\end{pmatrix}$. It has determinant $-1$, so it preserves area but reverses orientation; every point on $y=x$ is invariant.
• Common errors: multiplying in chronological rather than reverse matrix order, using $1/\det A$ without the adjugate, ignoring a negative determinant, and writing 'no inverse' without connecting it to a collapsed transformation or dependent equations.
• Exam technique: calculate the determinant early. It tells you whether inverse methods are legal, predicts the geometry, and often exposes arithmetic errors before they spread.

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