A 2 by 2 determinant is an area. The columns of the matrix are two arrows, and the determinant measures the parallelogram they span, with a sign that says which way round they turn. For 3 by 3 matrices the same number is a volume.

A photocopier set to 150 per cent enlarges every shape by the same factor, whatever the shape. A linear map does the same to areas, and its determinant is the factor, with a minus sign when the copy comes out mirrored.

The columns of $A=\begin{pmatrix}3&1\\1&2\end{pmatrix}$ are the arrows $\mathbf{a}_1=(3,1)$ and $\mathbf{a}_2=(1,2)$, and the dashed sides complete the parallelogram they span. Its area is $\det A=3\cdot2-1\cdot1=5$.

The same two arrows in the other order: now $\mathbf{a}_1=(1,2)$ and $\mathbf{a}_2=(3,1)$. The parallelogram is the same, but turning from to is now clockwise, and .

Here $\mathbf{a}_2=-2\mathbf{a}_1$: both columns lie on one line, the parallelogram is flat, and $\det (\begin{array}{cc}-2& 4\\ \end{array}$. The map squashes the whole plane onto that line.

For a $2\times2$ matrix $A$ with columns $\mathbf{a}_1$ and $\mathbf{a}_2$, the number is the area of the parallelogram with sides and . The sign is when turning from to is anticlockwise, the way turns to , and when it is clockwise. For a matrix, is the volume of the slanted box, the parallelepiped, whose edges are the three columns.

• The map $\mathbf{x}\mapsto A\mathbf{x}$ sends the unit square, with sides $\mathbf{e}_1$ and $\mathbf{e}_2$, to the parallelogram of the columns. So is the factor by which the map multiplies area, and it multiplies every region's area by that same factor.

The shear $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ keeps $\mathbf{e}_1$ and slides sideways to . The unit square, dashed on the left, becomes a leaning parallelogram with the same base and the same height, so its area is still — and .

Put the parallelogram with sides $(a,c)$ and $(b,d)$ inside the rectangle from the origin to its far corner $(a+b,c+d)$. What lies outside it is two pairs of right triangles and two small rectangles, and subtracting them leaves . This is the case where the four numbers are positive and the first side lies below the second; in the other positions different pieces are cut off, and the sign of records which way the sides turn.

The parallelogram fits inside the rectangle with one corner at the origin and the opposite one at its far vertex.

Two right triangles with legs $a$ and $c$ lie outside it: one under the side $(a,c)$, one above the side parallel to it.

Two right triangles with legs $b$ and $d$ lie beside the other pair of sides.

Two rectangles, $b$ wide and $c$ high, fill the two remaining corners.

Subtract everything outside the parallelogram from the rectangle; expanding the product leaves exactly the determinant.

Read the columns of the matrix as arrows from the origin. The parallelogram they span has area $|\det A|$.

So the parallelogram on $(3,1)$ and $(1,2)$ has area $5$: five times the unit square.

$\det A>0$ when the turn from the first column to the second is anticlockwise, like the turn from $\mathbf{e}_1$ to $\mathbf{e}_2$. when it is clockwise. Swapping the two columns keeps the parallelogram and reverses the turn, which is why it changes the sign.

Three columns in $\mathbb{R}^3$ are the edges of a slanted box. Its volume is $|\det A|$, and $\det A=0$ exactly when the box is flat: the three columns lie in one plane through the origin.

• $|\det A|$ is the area of the parallelogram spanned by the columns of a $2\times2$ matrix, and the volume of the box for a $3\times3$ one.
• The sign records orientation: positive for an anticlockwise turn from the first column to the second, negative for a mirrored picture.
• A linear map multiplies every area by $\mathrm{\mid }$.

Great work! You now know how to:

• read a 2 by 2 determinant as the area of a parallelogram;
• read its sign as the direction of a turn;
• use the determinant as the area factor of a linear map;
• read a 3 by 3 determinant as a volume, and a zero one as a flat box.