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Determinant of a matrix

The determinant of a square matrix is a number that provides a lot of useful information about the matrix.

Its definition is unfortunately not very intuitive. It is derived from abstract principles, laid out with the aim of satisfying a certain mathematical need. Therefore, before giving a definition of determinant, we explain what the mathematical need is.

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Understanding how linear transformations change volumes

Consider the linear space $S$ of all $2	imes 1$ real vectors. The space can be represented as a Cartesian plane. A vector $sin S$ can be represented as a point in the plane, whose x- and $y$-coordinate are the first and second entry of $s$ respectively.

Let $T_{1}subset S$ be a set of vectors such that the corresponding points in the Cartesian plane form a region whose area can be measured and is equal to $lpha _{1}$.

Now, take a $2	imes 2$ matrix A and define the set[eq1]

In other words, we use the square matrix A to linearly transform of all the vectors in the set $T_{1}$. The resulting linear transformations are in the new set $T_{2}$.

In the Cartesian space, the points of $T_{2}$ form a region whose area is equal to $lpha _{2}$.

We ask the following question: can we derive the area $lpha _{2}$ from $lpha _{1}$ and A?

It turns out that there is a number, called the determinant of A and denoted by [eq2], that satisfies[eq3]that is, when we multiply the area of the original region by the determinant, we obtain the area of the transformed region.

Importantly, the number depends only on A and not on $T_{1}$ and its area. The linear transformation defined by A transforms any region in a new region whose area is [eq4] times the area of the original region.

When we add one dimension and consider the space $S$ of $3	imes 1$ real vectors, we make obvious changes: we are no longer in the Cartesian plane, but in three-dimensional space; the matrix A used to perform linear transformations is a $3	imes 3$ matrix; the determinant is the scaling factor that allows to compute the volumes of the linearly transformed regions.

In K-dimensional spaces (for $K>3$), there are generalizations of the concept of volume, and linear transformations are defined by $K	imes K$ matrices, but the determinant keeps playing the same role: it acts as a scaling factor for the volume.

Signed volume

We have discussed volumes and how they are scaled by linear transformations, but we have omitted an important detail: when discussing determinants, we deal with signed volumes, that is, volumes that can be either positive or negative, depending on the orientation in space of the region whose volume is being measured. For this reason, also determinants can be positive or negative, depending on whether the linear transformation preserves or reverses the orientation of shapes.

As it can be difficult to get an intuitive grasp of the concept of signed volume, we are going to illustrate it with the following plots.

Six plots illustrating how the determinant of a linear tranformation helps to understand how the transformation changes the signed volume of a shape

The Cartesian plane in the upper left corner represents the original linear space. A set of points $T_{1}$ is depicted in blue (the circular arrow). The other five planes represent sets of points $T_{2}$ that have been obtained by applying different linear transformations to $T_{1}$ with different matrices A:

Many more linear transformations can be performed, for example, by combining the elementary transformations shown in the plot above. However, the principle remains the same: if the linear transformation associated to the matrix A does not change the orientation of the circular arrow, then [eq5]; if it changes the orientation, then [eq6] if the arrow is flattened (it loses one dimension), then [eq7].

Axiomatic approach

In the next section we are going to provide a definition of determinant that actually provides a way of calculating the determinant from the elements of A. That definition has been derived by mathematicians who took the following steps:

  1. they found some simple properties that are satisfied by the volumes of linearly transformed regions (e.g., if a linear transformation doubles the upper and lower edge of a rectangle and leaves its left and right edge unchanged, then the area of the rectangle doubles; as a consequence, the scaling factor of the linear transformation must be $2$);

  2. they imposed these properties as axioms that the determinant should satisfy;

  3. they proved that a number satisfying the axioms exists and is unique;

  4. they found a formula for calculating it, that can be used as its definition.

For more details about this axiomatic approach, you can refer to the beautiful treatment by Schneider and Barker (1989).

Definition of determinant

We are now ready to provide a formal definition of determinant.

Definition Let A be a $K	imes K$ matrix. Let $P$ be the set of all possible permutations of the first K natural numbers [eq8]The determinant of A, denoted by [eq9] or by [eq10], is[eq11]

In order to fully understand this definition you need to be familiar with the concepts of permutation and sign of a permutation.

A permutation $pi $ is an ordering of $1,ldots ,K$. The elements of the permutation are denoted by [eq12]. The number [eq13] is either 1 or $-1$ depending on the parity of the permutation (even or odd).

The product [eq14]is over K entries of the matrix A. For each row $k=1,ldots ,K$, we choose the entry located in column [eq15]. Note that there is exactly one chosen entry [eq16] in each column and row.

The sum $sum_{pi in P}$ is over the set $P$ of all possible permutations $pi $.

Determinant of a 2x2 matrix

Let us apply the definition to the case of a $2	imes 2$ matrix A.

There are two possible permutations of the set of the first two natural numbers:[eq17]

There are no inversions in $pi _{1}$, so its parity is even and[eq18]

There is one inversion in $pi _{2}$, so its parity is odd and[eq19]

Having established these facts, we can compute the determinant of A:[eq20]

Example Define the matrix[eq21]Its determinant is[eq22]

Determinant of a 3x3 matrix

Let us now tackle the case of a $3	imes 3$ matrix A.

There are six possible permutations of the set of the first three natural numbers. We report them below, together with the number of inversions and their sign (deriving them is left as an exercise - revise the lecture on the sign of a permutation if you find any difficulties):[eq23]

Therefore,[eq24]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Find the determinant of the matrix[eq25]

Solution

We can apply the formula for the determinant of a $2	imes 2$ matrix found above[eq26]

Exercise 2

The matrix used to perform the reflection in the lower left panel of the figure above is [eq27]Prove that [eq28].

Solution

We have [eq29]

References

Schneider, H., and Barker, G. P. (1989) Matrices and linear algebra, Dover Publications.

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