We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation.
Properties of Determinants of Matrices
We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about any row or column to make this computation. In this largely theoretical section, we will state and prove several more intriguing properties about determinants. Our main goal will be the two results in and , but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.
State the theorem that allows us to make yet another extension to our NMEx series of theorems.
What is amazing about the interaction between matrix multiplication and the determinant?
Determinants and Row Operations
We start easy with a straightforward theorem whose proof presages the style of subsequent proofs in this subsection.
So tells us the effect of the first row operation () on the determinant of a matrix. Here is the effect of the second row operation.
Let us go for understanding the effect of all three row operations. But first we need an intermediate result, but it is an easy one.
Now explain the third row operation. Here we go.
Is this what you expected? We could argue that the third row operation is the most popular, and yet it has no effect whatsoever on the determinant of a matrix! We can exploit this, along with our understanding of the other two row operations, to provide another approach to computing a determinant. We'll explain this in the context of an example.
As a final preparation for our two most important theorems about determinants, we prove a handful of facts about the interplay of row operations and matrix multiplication with elementary matrices with regard to the determinant. But first, a simple, but crucial, fact about the identity matrix.
If you asked someone with substantial experience working with matrices about the value of the determinant, they'd be likely to quote the following theorem as the first thing to come to mind.
For the case of $2\times 2$ matrices you might compare the application of with the combination of the results stated in and .
Example: Zero and nonzero determinant, Archetypes A and B
The coefficient matrix in has a zero determinant (check this!) while the coefficient matrix has a nonzero determinant (check this, too). These matrices are singular and nonsingular, respectively. This is exactly what says, and continues our list of contrasts between these two archetypes.
In we said singular matrices are a distinct minority. If you built a random matrix and took its determinant, how likely would it be that you got zero?
Since is an equivalence () we can expand on our growing list of equivalences about nonsingular matrices. The addition of the condition $\detname{A}\neq 0$ is one of the best motivations for learning about determinants.
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.