Determinant of a matrix

The determinant of a matrix $A$ is a value computed from the elements of a square matrix. Determinants are very useful mathematically, such as for finding inverses and eigenvalues and eigenvectors of a matrix and diagonalization, among other things. Determinants are denoted as $\det(A)$ or $|A|$ . A matrix that does not have a determinant of zero is called a nonsingular or nondegenerate matrix. Such a matrix will always be invertable and can be row-reduced to the identity matrix.

Computing the determinant

Determinants can only be computed if the matrix is square. If a matrix is 1×1, the determinant is simply equal to that element. If it is 2×2, such as

$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$

The determinant is equal to

$|A|=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$

For a 3×3 matrix, the determinant is taken by multiplying each element in a row or column by the determinant of the matrix left when removing all elements in the same row and column as the one we are multiplying by. For example:

$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=a\begin{vmatrix}e&f\\h&i\end{vmatrix}-b\begin{vmatrix}d&f\\g&i\end{vmatrix}+c\begin{vmatrix}d&e\\g&h\end{vmatrix}$

Any row or column will give the same result. Because of this, if any row or column is all zeros, the determinant will be zero. Note that the second term is negative. This is because the matrix can be thought of as a "checkerboard" of positives and negatives.

$\begin{vmatrix}+&-&+\\-&+&-\\+&-&+\end{vmatrix}$

Say we want to take the determinant around the second column. The expression would become

$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=-b\begin{vmatrix}d&f\\g&i\end{vmatrix}+e\begin{vmatrix}a&c\\g&i\end{vmatrix}-h\begin{vmatrix}a&c\\d&f\end{vmatrix}$

This can be extended to square matrices of any dimensions. Choosing a particular row or column can be a useful shortcut, as the more zeros there are the less terms have to be computed. For example, given a triangular or diagonal matrix, the determinant will be

$\begin{vmatrix}a&0&0&0\\b&c&0&0\\d&e&f&0\\g&h&i&j\end{vmatrix}=a\begin{vmatrix}c&0&0\\e&f&0\\h&i&j\end{vmatrix}=ac\begin{vmatrix}f&0\\i&j\end{vmatrix}=ac(fj-0i)=acfj$

In general, the determinant of any triangular matrix is

$|A|=a_{1,1}\cdots a_{n,n}=\prod_{i=1}^n a_{i,i}$

Properties of determinants

• Given a square matrix of dimension $n$ , $|mA|=m^n|A|$
• If two rows of columns of $A$ are swapped to produce $A$ , $|A'|=-|A|$
• If any two rows or columns of $A$ are identical, $|A| = 0$
• If any row or column of $A$ is multiplied by a scalar $m$ to produce $A$ , $|A'|=m|A|$
• Given a square matrix of dimension $n$ and its adjugate adj(A), $|\text{adj}(A)|={|A|}^{n-1}$
• $|A|I=A\ \text{adj}(A)$ , where $I$ is the identity matrix with the same dimensions as $A$ .
• If any non-zero scalar multiple of a row or column is added to another row or column of $A$ to produce $A$ , $|A|=|A'|$
• If any row or column has no non-zero entries, $|A| = 0$
• Given the matrices $A, B$ , $|AB| = |A||B|$
• $|A^T|=|A|$ , where $A^T$ is the transpose of $A$
• $|A^{-1}|=\frac1{|A|}$ , where $A^{-1}$ is the inverse of $A$ .
• $|A|=\prod_{i=0}^n\lambda_i=\lambda_1\cdots\lambda_n$ , for an $n \times n$ matrix where $\lambda_i$ is the $i$-th eigenvalue of $A$