Orthogonal matrices are really important when we calculate something called the determinant. They have special properties that make this process easier.

An orthogonal matrix, which we can call $A$, has a unique relationship: if we take the transpose of $A$ (which just means flipping it over its diagonal), it will equal its inverse (the matrix that, when multiplied with $A$, gives us the identity matrix). This relationship leads to a few important points about determinants.

First of all, the determinant of an orthogonal matrix can only be certain values. Specifically, we can say:

$$\text{det}(A) = \pm 1.$$

This means that when you use an orthogonal matrix to change space (like rotating or flipping it), the volume and the direction remain the same. This is helpful because instead of dealing with complicated calculations, you only need to check if the determinant is $1$ or $-1$.

Next, let's talk about what happens when we multiply two matrices together. For any two matrices $A$ and $B$, the determinant of their product works like this:

$$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B).$$

If either $A$ or $B$ is orthogonal, the overall result keeps the volume intact. So, when you multiply an orthogonal matrix with another matrix, it doesn’t change the volume, making it easier to evaluate larger transformations.

Also, orthogonal matrices are good at simplifying certain types of matrices. They help with a process called diagonalization, which makes finding determinants easier, especially for tricky matrices. The eigenvalues (which are special numbers related to the matrix) of an orthogonal matrix are located on something called the unit circle. Since their size is always $1$, this leads to straightforward calculations.

In conclusion, orthogonal matrices make it much easier to calculate determinants. They have fixed determinant values, helpful multiplication properties, and can simplify other matrices. This all helps us understand and work with linear transformations in higher dimensions.