Matrix determinant properties

The determinant has a lot of applications and special qualities that will not only help you to simplify many calculations but also to analyze some relationships between matrices.

In this topic, you will take a slightly more geometric and abstract approach to understand how the properties of the determinant arise. Also, you will learn two really important applications: the explicit calculation of the unique solution of a system of linear equations and a general expression for the inverse of a matrix!

A measure of distortion

You already know that a square matrix $A$ of size $2$ has associated a linear operator $L_A$ in $\mathbb{R}^2$. This operator converts the unit square whose sides are $e_1$ and $e_2$ into a parallelogram with sides $L_A(\mathbf{e_1})= A \, \mathbf{e_1} = A_1$ and $L_A(e_2)= A \, e_2 = A_2$:

This means that the area of the resulting parallelogram is precisely the determinant of $A$. So, the larger the area of this parallelogram, the greater the distortion made by $L_A$. For this reason, you can intuitively think of the determinant of a matrix as a rough measure of the size of the distortion generated by it.

The determinant as a function

In the previous section, you thought of $A$ in terms of its columns – both for its value in the canonical base and for the area of the parallelogram. This is why it's so natural to think of the determinant in terms of the columns rather than the matrix itself. In general, the determinant is a function that takes $n$ vectors in $\mathbb{R}^n$ and returns a real number. It's defined by:

$$\det: \underbrace{\mathbb{R}^n \times \dots \times \mathbb{R}^n}_{n-\text{times}} \rightarrow \mathbb{R} \text{ given by } \det(v_1, \dots, v_n) = |A| \text{ where } A=[v_1 | \dots | v_n]$$Clearly, the set $\underbrace{\mathbb{R}^n \times \dots \times \mathbb{R}^n}_{n-\text{times}}$ is the set of square matrices of size $n$. Making an analogy with the power of real numbers, this set is sometimes denoted by the symbol $\mathbb{R}^{n \times n}$. But what makes this function special, and what is the use of thinking about it in terms of vectors instead of matrices as you have done so far? Well, the determinant has a lot of special properties that you will learn about in this topic, but you can only establish and understand them with this new approach, so here you go!

First of all, the determinant is completely described as the only function with the following $3$ fundamental properties:

1. $\det(e_1, \dots, e_n) = 1$. Here, by definition, $\det(e_1, \dots, e_n) = |I|$ where $I$ is the identity matrix.
2. If among the vectors $v_1, \dots, v_n$ there are two equal, then $\det(v_1, \dots, v_n) = 0$. This property is called alternation.
3. If $\textcolor{teal}{j} \in \{1,\dots, n \}$, then fixing the vectors $v_1, \dots, v_{j-1}, v_{j+1}, \dots, v_{n}$, the function is linear with respect to column $\textcolor{teal}{j}$. That is, for every $u, w \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$:$$\det(v_1, \dots, v_{j-1}, \textcolor{teal}{u+w}, v_{j+1}, \dots, v_{n}) = \det(v_1, \dots, v_{j-1}, \textcolor{teal}{u}, v_{j+1}, \dots, v_{n}) + \det(v_1, \dots, v_{j-1}, \textcolor{teal}{w}, v_{j+1}, \dots, v_{n})$$$$\det(v_1, \dots, v_{j-1}, \textcolor{teal}{\lambda w}, v_{j+1}, \dots, v_{n}) = \textcolor{teal}{\lambda }\det(v_1, \dots, v_{j-1}, \textcolor{teal}{w}, v_{j+1}, \dots, v_{n})$$

The last property is known as multilinearity. Therefore, the determinant is the only alternating multilinear function on $\mathbb{R}^{n \times n}$ that sends the identity matrix to the number $1$. Now let's get to know its most useful properties!

The most useful properties

Hereafter let's consider the vectors $v_1, \dots, v_n$ in $\mathbb{R}^n$ and its matrix $A=[v_1 | \dots | v_n] \in \mathbb{R}^{n \times n}$.

The first property is a generalization of one you already know. If the vectors $v_1, \dots, v_n$ are linearly dependent, then $\det(v_1, \dots, v_n) = 0$

You already know that A is invertible if and only if its determinant is not zero. You can go a step further and calculate the determinant of $A^{-1}$:

Now let's take a look into geometry. If you extend the canonical vectors a bit, say by a factor of $2$, then the area of the new square doubles as well. But what about the area of the parallelogram? Does it double as well? No! Its area becomes $4$ times larger!

This is the perfectly natural thanks to the following property of the determinant:

Some other properties that come in handy and can save you more than one time are the following:

• The determinant of the transpose doesn't change: $$\det(A^T) = \det(A)$$
• The determinant of the product is the product of the determinants: $$\det(AB) = \det(A) \det(B)$$

Time for putting all this theory into practice.

Cramer's rule

Let's revisit systems of linear equations. Consider the system $Ax =b$ where $A$ is a square matrix of size $n$ and both $x$ and $b$ are vectors in $\mathbb{R}^n$.

You already know that the determinant helps to detect if the system has a unique solution. But there is a surprising result called Cramer's Rule that uses more determinants to explicitly calculate that unique solution!

Now let's see an example with the $3 \times 3$ system $Ax=b$ where:

$$A =\begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{bmatrix} \quad b=\begin{bmatrix} -3 \\ 0 \\ 1 \end{bmatrix}$$You can easily verify that $\det A = 2$ and:

• $\det A_{(1)} = \begin{vmatrix} \textcolor{teal}{-3} & 1 & 1 \\ \textcolor{teal}{0} & 2 & 4 \\ \textcolor{teal}{1} & 3 & 9 \end{vmatrix} = -16$
• $\det A_{(2)} = \begin{bmatrix} 1 & \textcolor{teal}{-3} & 1 \\ 1 & \textcolor{teal}{0} & 4 \\ 1 & \textcolor{teal}{1} & 9 \end{bmatrix} = 12$
• $\det A_{(2)} = \begin{bmatrix} 1 & 1 & \textcolor{teal}{-3} \\ 1 & 2 & \textcolor{teal}{0} \\ 1 & 3 & \textcolor{teal}{1} \end{bmatrix} = -2$

Thus, the unique solution is:

$$x = \frac{1}{\det A} \begin{bmatrix} \det A_{(1)} \\ \det A_{(2)} \\ \det A_{(3)} \end{bmatrix} = \frac{1}{2} \begin{bmatrix} -16 \\ 12 \\ -2 \end{bmatrix} = \begin{bmatrix} -8 \\ 6 \\ -1 \end{bmatrix}$$

Conclusion

Let's take some vectors $v_1, \dots, v_n$ in $\mathbb{R}^n$ and their matrix $A=[v_1 | \dots | v_n] \in \mathbb{R}^{n \times n}$.

• The determinant of $A$ is the area of the parallelogram into which $L_A$ converts to the unit square.

• The determinant is the only alternating multilinear function on $\mathbb{R}^{n \times n}$ that sends the identity matrix to the number $1$.

• $v_1, \dots, v_n$ are linearly independent if and only if $\det(v_1, \dots, v_n) \neq 0$.

• When $\det A \neq 0$, $\det(A^{-1}) = \frac{1}{\det A}$

• For any real number $\lambda$, it holds that $\det(\lambda A) = \lambda^n \det( A)$

• When a system of equations has a unique solution, you can use Cramer's rule to calculate it.