Properties of the Determinant

The determinant is a fundamental property of any square matrix. It is therefore important to know how the determinant is affected by various operations

Row Operations

This section outlines the effect that elementary row operations on a matrix have on the determinant

Row Switching

If $ B $ is a square matrix formed from swapping one row of $ A $ with another, then

$ \text{det}(B)=-\text{det}(A) $

If n number of row swaps have been made, then

$ \text{det}(B)=(-1)^n\text{det}(A) $

Row Scaling

If $ B $ is a square matrix formed from dividing one row of $ A $ by a number, k, then

$ \text{det}(B)=\frac{1}{k}\text{det}(A) $

Row Addition

If $ B $ is a square matrix formed from adding one row of $ A $ to another, then

$ \text{det}(B)=\text{det}(A) $

Calculating Determinant from RREF

By combining the previous three properties and tracing the math you use to get to the reduced row echelon form you can easily calculate the determinant. Just keep track of how many row swaps and scalings you make. If you use n row swaps and p divisions then

$ \text{det}(A)=(-1)^n*\text{det}(\text{rref}(A))*\prod_{i=1}^{p}\frac{1}{k_i} $

where $ \frac{1}{k_i} $ is the ith scaling factor

One thing to note is that if you go all the way to RREF, then the determinant of the reduced row echelon form must either be one or zero because the reduced row echelon form is either the identity or has a row of zeros. From this we get our next property

Invertibility

A is invertible if and only if

$ \text{det}(A)\ne 0 $

Matrix Multiplication

This is best explained simply with an equation.

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

And since the right hand side is simply multiplication of scalars, it is commutative, and therefore we can show that

$ \begin{align} \text{det}(AB)&=&\text{det}(A)\text{det}(B)\\ &=&\text{det}(A)\text{det}(B)\\ &=&=\text{det}(BA)\\\end{align} $

This is an incredible thing.