C5 - Determinants in two and three dimensions

Expanding determinants

Can a matrix send two different inputs to the same output? For $\begin{pmatrix}1&2\\2&4\end{pmatrix}$, the inputs $\begin{pmatrix}2\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$ both give $\begin{pmatrix}2\\4\end{pmatrix}$. A single number, the determinant, will detect this loss of information.

For the opening matrix the determinant is $1(4)-2(2)=0$. First practise the calculation before interpreting it.

Change only the bottom-right entry from 3 to $-3$. The second diagonal product stays the same, but the first changes sign.

For order 3, reduce the calculation to determinants of order 2. The minor of an entry is the determinant left after deleting its row and column. Its cofactor is that minor multiplied by the sign in this pattern:

Multiply each entry of one row by its cofactor and add. This is cofactor expansion. Along the top row,

You may instead expand down any column or along any row, using its own cofactor signs. Choose a row or column containing zeros to reduce the work. The expansion rule is not in the formula booklet.

Keep the matrix unchanged and change only the expansion row. This isolates the cofactor sign attached to a different position.

AQA's matrix questions often embed determinant work inside an inverse calculation. Keeping the minor, its cofactor sign and its multiplying entry separate makes errors easier to find.

Use a fresh matrix and choose an efficient expansion.

Zero determinant and collapse

Think of $\mathbf{M}\begin{pmatrix}x\\y\end{pmatrix}$ as a rule sending an input point to an output point. Its two columns are the outputs of $\begin{pmatrix}1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$, and every output is a linear combination of those columns.

For $\mathbf{M}=\begin{pmatrix}1&2\\2&4\end{pmatrix}$, the second column is twice the first. Every output is therefore

All outputs lie on the line $Y=2X$. A whole square of inputs collapses onto a line; its image has area zero. The two distinct inputs in the opening question share one output because their contributions combine to the same value of $x+2y$.

[DIAGRAM: asset_name: A square collapses onto a line; asset_slug: aqa7367-zero-determinant; caption: The matrix with columns (1,2) and (2,4) sends a unit square onto a line segment; description: Two coordinate plots show the unit square with vertices (0,0), (1,0), (1,1), (0,1), and its four labelled images (0,0), (1,2), (3,6), (2,4). The image points all lie on Y equals 2X, so the image has zero area.]

For a $2\times2$ matrix, a zero determinant means its columns lie along a single line, or are both zero. The image has at most one independent direction: a line or a point. A non-zero determinant means the two columns retain two independent directions.

In three dimensions, three independent column directions span space. If a $3\times3$ matrix has determinant zero, those directions are dependent, so a solid can collapse into a plane, a line or a point. For example,

This matrix has determinant zero and sends every point to the plane $z=0$. Changing the input's third coordinate no longer changes the output. The $3\times3$ zero matrix collapses everything to the origin.

The magnitude of a determinant gives the area factor in two dimensions or volume factor in three dimensions. At zero, that area or volume vanishes. A non-zero determinant also carries orientation information; you will develop signed scale factors with successive transformations.

Distinguish a collapse into a plane from a collapse to the origin.

Quick Recap

• A determinant is a scalar associated with a square matrix; for order 2 it is $ad-bc$.
• For order 3, expand along one row or column, multiplying entries by signed minors.
• A zero determinant signals collapse into fewer independent dimensions, rather than requiring every output to be zero.

Yes: distinct inputs can give the same output. The opening matrix loses the distinction between inputs whose value of $x+2y$ is the same.