4.3 Some applications of determinants
In this section we will collect a few applications of determinants. Let us also note that everything we’ve done about determinants has not used specifically that we were dealing with reals, so all the above results hold if one wishes to replace \(\mathbb{R}\) by, say \(\mathbb{C}\) or \(\mathbb Z\). In particular, note that the determinant of an integer square matrix is an integer.
4.3.1 Determinants and systems of equations
Recall that system of \(m\) linear equations in \(n\) unknowns can be written in the form \[Ax=b,\] where \(A\in M_{m,n}(\mathbb{R})\) and \(b\in \mathbb{R}^m\), and \(x\in \mathbb{R}^n\) is the vector of the unknowns. If \(m=n\), i.e., the system has as many equations as unknowns, then \(A\) is a square matrix and so we can ask if it is invertible.
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We know that \(A\) is invertible if and only if \(\det A\neq 0\), and then we find \[x=A^{-1}b.\] So \(\det A\neq 0\) means the system has a unique solution.
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In this case we could have either infinitely many solutions or no solutions
If \(\det A=0\) then the row echelon form of \(A\) must have at least one row of zeros. Then whether the system has no solutions or infinitely many solutions will depend on the row echelon form of the augmented matrix \((A \,b)\). Then by Theorem 3.39 this has the same number of zero rows then there will be infinitely many solutions; if it has fewer then there will be no solutions.
If \(\det A\neq 0\) one can go even further and use the determinant to compute an inverse and the unique solution to \(Ax=b\).
So to calculate the adjugate we find the signed minors of the original matrix, put these in as the entries in the corresponding place in the matrix, and then take the transpose.
This matrix is useful. Firstly, it gives us an explicit formula for \(A^{-1}\).
Let \(A\in M_{n}(\mathbb{R})\), with \(\det A\neq 0\). Then \[A^{-1}=\frac{1}{\det A} \operatorname{adj}A.\]
The following related result is called Cramer’s rule, and gives the explicit formula for solutions of systems of linear equations.
Let \(A\in M_{n}(\mathbb{R})\), with \(\det A\neq 0\). Let \(b\in \mathbb{R}^n\) and \(A_j\) be the matrix obtained from \(A\) by replacing its \(j\)th column by \(b\). Then the unique solution \(x=(x_1, x_2,\cdots ,x_n)\) to \(Ax=b\) is given by \[x_j=\frac{\det A_j}{\det A} , j=1,2, \cdots, n .\]
Both results can be proven by playing around with Laplace expansion and other basic properties of determinants. These results are mainly of theoretical use since in practice Gaussian/Gauss-Jordan elimination is more efficient at solving systems of equations or finding matrix inverses as it is a more scalable process.
4.3.2 The cross product
The determinant can be used to define the cross product of two vectors in \(\mathbb{R}^3\), which will be another vector in \(\mathbb{R}^3\). If we recall Laplace expansion in the first row for a \(3\times 3\) matrix, \[\begin{equation} \det A=a_{11} A_{11}+ a_{12} A_{12}+a_{13} A_{13}, \tag{4.7}\end{equation}\] then we can interpret this as the dot product between the first row-vector of \(A\) and the vector \(( A_{11}, A_{12}, A_{13})\) whose components are the signed minors associated with the first column. If we denote the first column by \(z=(z_1,z_2,z_3)\) and the second and third by \(x=(x_1,x_2.x_3)\) and \(y=(y_1,y_2,y_3)\), then the above formula reads \[\det\begin{pmatrix}z_1 & x_1 & y_1\\ z_2& x_2 & y_2\\ z_3 & x_3 & y_3\end{pmatrix}=z_1(x_2y_3-x_3y_2)+z_2(x_3y_1-x_1y_3)+z_3(x_1 y_2-x_2y_1) ,\] and we therefore use this to define the cross product.
The formula (4.7) then becomes \[\begin{equation} \det\begin{pmatrix}x_1 & x_2 & x_3\\ y_1& y_2 & y_3\\ z_1 & z_2 & z_3\end{pmatrix}= \det\begin{pmatrix}z_1 & x_1 & y_1\\ z_2& x_2 & y_2\\ z_3 & x_3 & y_3\end{pmatrix}=z\cdot (x\times y). \tag{4.8}\end{equation}\]
The cross product, and notions derived from it, appear in many applications: mechanics, vector calculus, geometry. Let us collect now a few properties.
The cross product is a map \(\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R}^3\) which satisfies the following properties for any \(x, y, z\in \mathbb{R}^3\) and \(\alpha, \beta \in \mathbb{R}\):
Antisymmetry: \(y\times x=-x\times y\) and \(x\times x=\mathbf{0}\).
Bilinearity: \((\alpha x+\beta y)\times z=\alpha (x\times z)+\beta (y\times z)\).
\(x\cdot (x\times y)=y\cdot (x\times y)=0\).
\(\lVert x\times y\rVert^2=\lVert x\rVert^2\lVert y\rVert^2-(x\cdot y)^2\).
\(x\times (y\times z)=(x\cdot z)y-(x\cdot y)z\).
We will leave this as an exercise. The first three properties follow easily from the relation (4.8) and properties of the determinant, and the remaining two can be verified by direct, although not so easy computations.
Property (iii) means that \(x\times y\) is orthogonal to the plane spanned by \(x\) and \(y\), and (iv) gives us the length as \[\begin{equation} \lVert x\times y\rVert^2=\lVert x\rVert^2\lVert y\rVert^2 \lvert\sin \theta\rvert^2, \tag{4.9}\end{equation}\] where \(\theta\) is the angle between \(x\) and \(y\) (since \((x\cdot y)^2=\lVert x\rVert^2\lVert y\rVert^2\cos ^2\theta\)) . Let \(n\) be the unit vector (i.e., \(\lVert n\rVert=1\)) orthogonal to \(x\) and \(y\) chosen according to the right hand rule: if \(x\) points in the direction of the thumb, \(y\) in the direction of the index finger, then \(n\) points in the direction of the middle finger. For example, if \(x=e_1\), \(y=e_2\) then \(n=e_3\), whereas \(x=e_2\), \(y=e_1\) gives \(n=-e_3\). Then we have the following result.
The cross product satisfies \[x\times y=\lVert x\rVert\lVert y\rVert\lvert\sin\theta\rvert n.\]
Physicists often take this as the definition of the cross product. We omit the proof.
Property (v) of Theorem (4.7) implies that the cross product is not associative, i.e., in general \((x\times y)\times z\neq x\times(y\times z)\). Instead the so called Jacobi identity holds (which can be verified directly using (v) in Theorem 4.33 ): \[x\times (y\times z)+y\times (z\times x)+z\times (x\times y)=\mathbf{0}.\]
In summary, we have the following geometric interpretations,:
\(x\cdot x\) is the square of the length of the vector \(x\);
\(\lVert x\times y\rVert\) is the area of the parallelogram spanned by \(x\) and \(y\);
\(x\cdot (y\times z)\) is the oriented volume of the parallelepiped spanned by \(x, y,z\).
More details on all this are on problem sheets. Finally, note that although the dot product makes sense for pairs of vectors in any \(\mathbb{R}^n\), the cross product is defined for vectors in \(\mathbb{R}^3\) only.
As we have seen, the determinant function has a range of applications, and as we continue through the course we will encounter some additional ones.
Having explored matrices and determinants, we are now ready to explore the concept of vector spaces and their properties in more detail.