CP 3.5 - Determinants and scale factors

The determinant as signed area

Can a transformation move corners of a square while leaving its area unchanged? Consider the matrix $\mathbf{S} = \begin{pmatrix}1&2\\0&1\end{pmatrix}$. It fixes $(0,0)$ and $(1,0)$, but moves the upper corners to $(2,1)$ and $(3,1)$. The square becomes a sloping parallelogram. Its base and perpendicular height are still both $1$, so its area is still $1$.

The determinant measures this change of area without requiring you to draw every transformed shape. To see where its formula comes from, let the columns be $(a,c)$ and $(b,d)$. The image of the unit square has vertices $(0,0)$, $(a,c)$, $(a+b,c+d)$ and $(b,d)$. Subtracting the triangles outside this parallelogram, or using the coordinate area formula, gives the signed area $ad-bc$.

[DIAGRAM: asset_name: A unit square and its sheared image; asset_slug: 9fm0-cp35-shear-area; caption: The square and its sheared image both have base 1 and perpendicular height 1; description: Two coordinate panels show the unit square with vertices (0,0), (1,0), (1,1), (0,1), then its image with vertices (0,0), (1,0), (3,1), (2,1) under S with rows (1,2) and (0,1). Blue and lavender arrows show the images of the first and second unit vectors. Each panel labels area 1 and perpendicular height 1.]

Orientation means the order in which you travel round corresponding labelled vertices. If $O,A,B$ are anticlockwise and their images are clockwise, orientation is reversed. A positive determinant preserves orientation; a negative determinant reverses it. A rotation through $180^\circ$ preserves orientation even though every non-zero position vector points the opposite way afterwards.

For $\mathbf{S}$, replacing $2$ by any real number $t$ still gives determinant $1$. You can shear the top of a square as far sideways as you please without changing its area. Lengths and angles change; area survives. This is why determinant $1$ does not mean “a rotation”.

The next check separates an area statement from a statement about lengths.

Calculating two by two determinants

For a two by two matrix, multiply down the main diagonal and subtract the product on the other diagonal. Keep the subtraction visible when entries are negative.

The vertical bars are an alternative notation for the determinant, so $\begin{vmatrix}a&b\\c&d\end{vmatrix}$ means $ad-bc$, not a matrix with different brackets.

Worked example 2 changes the sign of one entry. This time subtracting a negative product is the decision to watch.

An entry and the formula's subtraction can supply separate minus signs.

Change the sign of the other off-diagonal entry in your own calculation.

Calculating three by three determinants

A three by three determinant reduces to three two by two determinants. The alternating signs do the bookkeeping; choosing a row containing a zero reduces the arithmetic.

Deleting the row and column of an entry leaves a two by two matrix. Its determinant is the entry's minor. For expansion along the top row, the signs are plus, minus, plus.

Each coefficient multiplies the minor obtained by deleting its own row and column. You may expand along any row or column, with signs from the checkerboard $\begin{pmatrix}+&-&+\\-&+&-\\+&-&+\end{pmatrix}$. These signed minors are called cofactors; they will reappear when you find inverses.

Worked example 4 changes only the top-right entry, from zero to a negative number. All three terms now contribute.

The common failure is to handle the middle term as positive, especially when its coefficient is already negative.

One more change to the top-right entry gives a closely matched check.

Area, volume and orientation

Once you have the determinant, interpreting it is often shorter than calculating it. Keep the signed determinant and the positive scale factor as separate statements.

In three dimensions, orientation distinguishes right-handed and left-handed arrangements of three independent directions. A reflection in a coordinate plane reverses orientation. A rotation preserves it. A three-dimensional determinant scales volume, not the area of every surface: a flattened box can have zero volume while retaining a non-zero face area.

Worked example 6 changes the dimension. The same interpretation now concerns volume, and the determinant comes from three directions.

A negative determinant changes orientation without making a geometric volume negative.

For successive transformations, the scale factors multiply. In determinant notation, $\det(\mathbf{A}\mathbf{B})=\det\mathbf{A}\det\mathbf{B}$; this result is not in the formula booklet. Two negative determinants therefore give a positive determinant: reversing orientation twice restores it. For an enlargement with length scale factor $k$, the determinant is $k^2$ in two dimensions and $k^3$ in three dimensions. A determinant is rarely a length scale factor.

The opening shear now has a numerical explanation: its determinant is $1$, so every area, not just the square's area, is preserved.

Quick Recap

• For a two by two matrix, $\det\mathbf{M}=ad-bc$.
• For a three by three matrix, expand along a row or column using signed two by two minors.
• $|\det\mathbf{M}|$ is the area factor in two dimensions and the volume factor in three dimensions.
• A positive determinant preserves orientation; a negative determinant reverses it; a zero determinant collapses dimension.

Yes, the opening shear moves corners without changing area: $\det\begin{pmatrix}1&2\\0&1\end{pmatrix}=1$.