What are all the uses of the determinant?

Question

I've learned how to calculate the determinant but what is the determinant used for? So far, I only know that there is no inverse if the determinant is 0.

Answer 1

Determinants can be used to see if a system of $n$ linear equations in $n$ variables has a unique solution. This is useful for homework problems and the like, when the relevant computations can be performed exactly.

However, when solving real numerical problems, the determinant is rarely used, as it is a very poor indicator of how well you can solve a system of equations, and furthermore, it is typically very expensive to compute directly. Other quantities (such as singular values) provide better indications of 'solvability', and other techniques (Gaussian elimination, QR decompositions, etc.) are better for solving systems of equations.

The determinant also gives the (signed) volume of the parallelepiped whose edges are the rows (or columns) of a matrix. I find this interpretation to be the most intuitive, and many standard results for determinants can be understood using this viewpoint. (However, I have rarely had a practical need to compute volumes using determinants.) The volume interpretation is often useful when computing multidimensional integrals ('change of variables'). It is also useful for understanding (or defining) the 'cross product' in physics or mechanics.

The determinant is a very useful theoretical tool, whose applications extend well beyond matrices of real or complex numbers. However this may not be apparent at the calculus level.

Answer 2

1. It allows you to evaluate cross products and find the general equation of the plane if given $3$ points. For instance: $A(1,1,0),\, B =(1,0,1),\,C=(0,1,2)$ $$B-A=(1,0,1)-(1,1,0)=(0,-1,1)$$ and $$C-A=(0,1,2)-(1,1,0)=(-1,0,2)$$ You now use the cross-product of $$(B-A)\times(C-A)=\begin{bmatrix}i & j & k \\0 & -1 & 1 \\-1 & 0 & 2 \end{bmatrix}=(-2,-1,-1)= \vec{n}$$ $i$, $j$ and $k$ are unit vectors that are directed along the $x$, $y$ and $z$ axis respectively. $\vec{n}$ is known as the normal vector $\vec{n}$ and is perpendicular to the equation of the plane.

   2. For square matrices formulae such as $$\det(AB) = \det (A) \det (B)$$ and $$\det (A^{-1}) = \frac 1 {\det (A)}$$ are applicable

   3. Allows you to check if your matrix has an inverse: $$\det=\begin{vmatrix}0 & 0 & 2 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 3 & 0\\ 2 & 1 & 5 & -3 \end{vmatrix}$$

$$=0\times\begin{vmatrix} 0 & 0 & 1 \\ -1 & 3 & 0 \\ 1 & 5 & -3 \end{vmatrix}+0\times\begin{vmatrix} 1 & 0 & 1 \\ 0 & 3 & 0 \\ 2 & 5 & -3 \end{vmatrix}+2\times\begin{vmatrix} 1 & 0 & 1 \\ 0 & -1 & 0 \\ 2 & 1 & -3 \end{vmatrix}+0\times\begin{vmatrix} 1 & 0 & 0 \\ 0 & -1 & 3 \\ 2 & 1 & 5 \end{vmatrix}$$$$=2\times\begin{vmatrix} 1 & 0 & 1 \\ 0 & -1 & 0 \\ 2 & 1 & -3 \end{vmatrix}$$ $$=2\left( 1\times\begin{vmatrix}-1 & 0 \\ 1 & -3 \\ \end{vmatrix}+0\times\begin{vmatrix}0 & 0 \\ 2 & -3 \\ \end{vmatrix}+ 1\times\begin{vmatrix}0 & -1 \\ 2 & 1 \\ \end{vmatrix}\right)$$$$=2\times \left(1\times(3-0)+1\times(0--2)\right)= 2(3+2)=10 \ne 0$$ So your matrix is invertible (or has an inverse).

Answer 3

As pointed out it can have several applications. I'll give a particular application in coding theory, specifically in the design of space-time codes, which are codes used in wireless systems with multiple transmitter and receiver antennas.

These space-time codes can be viewed as matrices $X$, where one dimension represents space (i.e., number of different antennas) and other dimension represents time (assume both equal for now). The elements of the matrix are taken from a complex field $\mathbb{C}$. A code book $\mathcal{C}$ is a collection of such code matrices. For such space-time codes a key criteria of designing a good code is to design matrices having large minimum determinant of the difference of any two code matrices in this code book i.e. define $\delta=\min_{X_1,X_2 \in \mathcal{C}} |\det(X_1-X_2)|$. The larger the value of $\delta$ the better and more efficient is the code (efficient in terms of power utilization and data rate). So this is another interesting area where computing determinant is of practical use.