---
title: Determinant
tags: Linear algebra
GA: G-77TT93X4N1
---

# Determinant

> Determinant is a function that maps a $n\times n$ matrix to a real value.

## Geometric meaning

Consider a matrix

$$
A = \begin{bmatrix}
{\bf v}_1 & {\bf v}_2 & \cdots & {\bf v}_n
\end{bmatrix},
$$
where $c_i\in\mathbb{R}^n$ are column vectors. Determinant is the "volume" of the $n$-dimensional parallelepiped that is constructed by $c_i$'s.

---

## Notation

$$
det(A) = |A| = D({\bf v}_1, \cdots, {\bf v}_n)
$$

---

## Definition

1. Determinant of identity matrix is one
$$
det(I_n) = 1
$$
2. Antisymmetry
$$
D(\cdots, {\bf v}_j, \cdots, {\bf v}_k,\cdots) = -D( \cdots, {\bf v}_k, \cdots, {\bf v}_j,\cdots)
$$
3. Linearity in each argument:
$$
D(\cdots, \alpha{\bf v}_k, \cdots) = \alpha D( \cdots, {\bf v}_k, \cdots)
$$
and
$$
D(\cdots, {\bf u}_k+{\bf v}_k, \cdots) = D( \cdots, {\bf u}_k, \cdots) + D( \cdots, {\bf v}_k, \cdots)
$$

---

## Properties

> The proof of these properties are given in [Determinant 2](https://hackmd.io/@NYCUAM/LA_determinant_2).

1. Two equal columns, then determinant equals to zero.
2. A column of zero, then determinant equals to zero.
3. Preservation under “column replacement”
$$
D(\cdots, {\bf v}_j+\alpha{\bf v}_k, \cdots, {\bf v}_k,\cdots) = D( \cdots, {\bf v}_j, \cdots, {\bf v}_k,\cdots).
$$
4. If $A$ is singular, then $\text{det}(A) =0$.
5. Determinant of upper triangular matrix is the product of its diagonal elements, i.e.,
$$
\left|\begin{matrix}
d_1 & * & \cdots & *\\
0 & d_2 & \cdots & *\\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & d_n \\
\end{matrix}\right| = d_1d_2\cdots d_n.
$$
* Remark: Same conclusion applied to lower triangular matrices.
6. $\text{det}(AB) = \text{det}(A)\text{det}(B)$.
7. $\text{det}(A^T) = \text{det}(A)$.

## References
* [Linear Algebra Done Wrong: Ch2](https://www.math.brown.edu/streil/papers/LADW/LADW.html)