Determinants

The determinant of a matrix is a scalar value just like the rank that says something about the matrix. The determinant is only defined for square matrices and is denoted as $det(A)$ or $|A|$ for the matrix $A \in \mathbb{R}^{n \times n}$. The determinant of a matrix is $0$ if the matrix is singular. So in other words if the determinant of a matrix is $0$, then the matrix can not be inverted and the rank of the matrix is less than $n$ where $n$ is the number of rows and columns of the matrix. For this reason, the determinant is a useful tool in linear algebra as it can be used to check if a matrix is invertible before attempting to invert it.

Calculating Determinants

Determinant of a 2x2 Matrix

Info

From the 3Blue1Brown video on determinants, the determinant of a matrix can be thought of as the scaling factor of the matrix when it is used as a transformation matrix. Imagine the normalized basis of a vector space forming a square or a cube with area or volume of 1. When the transformation matrix is applied to the basis, the determinant of the matrix tells us how much the area or volume of the basis has been scaled by. If the determinant is 0, then the area or volume of the basis has been squinched down to 0 because the matrix has squished the basis down to a lower dimension. If the determinant is negative, then the area or volume of the basis has been flipped.

Determinant of a 3x3 Matrix

Rule of Sarrus is the pretty pattern?

Determinant of a 4x4 Matrix

Laplace Expansion

general formula for calculating the determinant of an n x n matrix

works recursivly for n x n matrices also for n=2.

Determinant of a Triangular Matrix

The determinant of a triangular matrix is the product of the diagonal elements of the matrix.

Characteristic Polynomial

Allows us to find the eigenvalues of a matrix and represent a matrix as a polynomial

Effect of Operations on Determinants

Transpose

When transposing a matrix the determinant stays the same. This is because all the transpose only changed the order in which the elements are multiplied with each other and added together. Multiplication and addition are commutative operations so the order does not matter as long as the grouping of the elements stays the same.

Swapping Rows

Each time two rows are swapped the determinant of the matrix changes sign.

Adding Multiples of Rows

When a multiple of one row is added to another row the determinant of the matrix stays the same. The extra terms cancel each other out when calculating the determinant.

Row Scaling

When a row is scaled by a factor $k$ the determinant of the matrix is scaled by the same factor $k$.

Therefore if the entire matrix is scaled by a factor $k$ the determinant of the matrix is scaled by $k^n$ where $n$ is the number of rows or columns of the square matrix.