Determinants

Geometric Intuition §

One good way of measuring linear transformations is with how the area of a given region changes. You can think of the determinant of a matrix as a measure of the linear transformation the matrix represents.

When a linear transformation is applied to the space, every region in the space changes area by the same factor, called the determinant. For example, if a transformation increases every area by 3, its determinant 3

If a linear transformation transforms the area of a region into 0, then the determinant must be 0. This is a special case and has applications, such as when calculating eigenvectors and eigenvalues. We can visualize this linear transformation as squishing a grid into a lower dimension

If the orientation of space is inverted, then determinant is negative. You can use RH/LH rule to verify change in orientation.

Above properties are for 2-D but apply to 3D as well.

Calculation §

The determinant of a $2\times 2$ matrix is calculated as follows:

$$\det \left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=ad-bc$$

Laplace’s Expansion §

For larger matrices, such as

$$A=\left(\begin{array}{ccccc}2& 0& 0& 1& 3\\ 0& 4& 0& 5& 0\\ 1& 0& -1& 0& 0\\ 0& 0& 0& 2& 1\\ 4& 1& 0& 0& -2\end{array}\right)$$

we use Laplace’s Expansion. Laplace’s Expansion states that the definition of the determinant of a matrix is

$$\det A=\sum _{i=1}^n(-1{)}^{i+1}{a}_{i_1}$$ $$\begin{array}{rl}\det (A)& =1\cdot \det (\left(\begin{array}{cccc}0& 0& 1& 3\\ 4& 0& 5& 0\\ 0& 0& 2& 1\\ 1& 0& 0& -2\end{array}\right))-1\cdot \det (\left(\begin{array}{cccc}2& 0& 1& 3\\ 0& 4& 5& 0\\ 0& 0& 2& 1\\ 4& 1& 0& -2\end{array}\right))\\ & =0-2\det \left(\begin{array}{ccc}2& 0& 3\\ 0& 4& 0\\ 4& 1& -2\end{array}\right)+\det \left(\begin{array}{ccc}2& 0& 1\\ 0& 4& 5\\ 4& 1& 0\end{array}\right)\\ & =-2\cdot 4\det \left(\begin{array}{cc}2& 3\\ 4& -2\end{array}\right)+2\det \left(\begin{array}{cc}4& 5\\ 1& 0\end{array}\right)+\det \left(\begin{array}{cc}0& 4\\ 4& 1\end{array}\right)\\ & =128-10+-16\\ & =102\end{array}$$

Triangle Method §

If we have a matrix with an upper or lower triangle (upper triangle shown below)

$$A\to \left(\begin{array}{ccccc}2& 0& 0& 1& 3\\ 0& 4& 0& 5& 0\\ 0& 0& -1& -\frac{1}{2}& -\frac{3}{2}\\ 0& 0& 0& 2& 1\\ 0& 0& 0& 0& -\frac{51}{8}\end{array}\right)$$

Then, multiplying along the diagonal and by the determinants of any elementary matrices used in converting to row echelon form:

$$\det A=102$$

This is just an extension of Laplace’s Expansion.

Determinant of Matrix Product §

The determinant the product of two matrices is just the product of their determinants.

$$\det (AB)=\det A\cdot \det B$$

Proof §

To prove it, we must start with proving the above property for elementary matrices.

Row Swapping §

The formula for the determinant of a row-swapping elementary matrix $\mathbf{E}$ by a matrix $\mathbf{A}$ is

$$-\det \mathbf{A}$$

Our lord and savior Chris Ge provides both an informal and formal proof. Since I hate collared shirts, the informal proof is as follows:

1. Swapping adjacent rows -> determinant is multiplied by -1 (Laplace’s Expansion)
2. Swapping two non-adjacent rows $i$ and $j$ is the same as making $2(j-i)-1$ swaps -> determinant is multiplied by -1.

Row Multiplication §

Given an elementary matrix $\mathbf{E}$ that multiplies a row of a matrix $\mathbf{A}$ by a factor $c$, the determinant of $\mathbf{EA}$ is

$$c\det \mathbf{A}$$

or equivalently

$$\det \mathbf{C}\det \mathbf{A}$$

We know that the determinant of $\mathbf{C}$ is just $c$ by the Triangle Method.

The proof for this formula is pretty intuitive; if you use Laplace’s Expansion along the row of $\mathbf{A}$ that was scaled by the factor $c$, then the determinant is naturally going to be multiplied by $\frac{1}{c}$.

Row Addition/Subtraction §

Given an elementary matrix $\mathbf{E}$ that performs row addition within a matrix $\mathbf{A}$, the determinant of $\mathbf{EA}$ is just

$$\det \mathbf{A}$$

or equivalently

$$\det \mathbf{E}\det \mathbf{A}$$

This is because the determinant, according to Laplace’s Expansion, is a linear function of the rows and columns of the matrix. Therefore, adding rows to each other means that they cancel out eventually.

Here’s an example. Let

$$\mathbf{A}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$$

The determinant is

$$\det \mathbf{A}=a_{11}(a_{22}{a}_{33}-a_{23}{a}_{32})-a_{12}(a_{21}{a}_{33}-a_{23}{a}_{31})+a_{13}(a_{21}{a}_{32}-a_{22}{a}_{31})$$

Now, if we add two of $R_1$ to $R_2$,

$$\mathbf{A}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}+2a_{11}& a_{22}+2a_{12}& a_{23}+2a_{13}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$$

The determinant is

$$\begin{array}{rl}\det \mathbf{A}& =a_{11}((a_{22}+2a_{12})a_{33}-(a_{23}+2a_{13})a_{32})\\ & -a_{12}((a_{21}+2a_{11})a_{33}-(a_{23}+2a_{13})a_{31})\\ & +a_{13}((a_{21}+2a_{11})a_{32}-(a_{22}+2a_{13})a_{31})\\ & =a_{11}(a_{22}{a}_{33}-a_{23}{a}_{32})-a_{12}(a_{21}{a}_{33}-a_{23}{a}_{31})+a_{13}(a_{21}{a}_{32}-a_{22}{a}_{31})\end{array}$$

Thus ending our proof.

Combining Elementary Matrices §

Now that we have our above lemmas, we can just combine them all to say that for any matrices $\mathbf{A}$ and $\mathbf{B}$, as long as one of them are invertible,

$$\det \mathbf{AB}=\det \mathbf{A}\det \mathbf{B}$$

This is because invertible matrices can be written as the product of a series of elementary matrices, allowing us to expand out the determinant of the product as a product of many determinants of elementary matrices.

$$\begin{array}{rl}\det \mathbf{AB}& =\det ({\mathbf{E}}_1{\mathbf{E}}_2\dots {\mathbf{E}}_n\mathbf{B})\\ & =\det {\mathbf{E}}_1\det {\mathbf{E}}_2\dots \det {\mathbf{E}}_n\det \mathbf{B}\end{array}$$

Properties §

1. If $A$ is invertible

$$\det A^{-1}=\frac{1}{\det A}$$

2. If one row of $A$ is a multiple of another row, then

$$\det A=0$$

3. If A has a row or column of all zeroes, then

$$\det A=0$$

4. If $\det A=0$, then $A$ is not invertible