Determinant

Defined only for square matrices. Denoted by $\lvert A\rvert$. $$

For 2x2

$$\lvert A\rvert= \Bigg\lvert{ \, \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \, } \Bigg\rvert = a_{11}a_{22} - a_{12}a_{21}$$

For higher order

Minor of an element

Suppose $A=(a_{ij})$. $$

Minor of an element $a_{ij}$, is the matrix obtained by deleting $i^{\text{th}}$ row and $j^{\text{th}}$ column of $A$. Denoted by $M_{ij}$.

Co-factor of an element

Suppose $A=(a_{ij})$. $$

Co-factor of an element $a_{ij}$, is defined as (commonly denoted as $A_{ij}$):

$$A_{ij} = (−1)^{i+j}\,\lvert M_{ij}\rvert$$

Definition

If $A = (a_{ij})_{n\times n}$ then the determinant of $A$ is defined by:

$$\lvert A\rvert= \sum_{j=1}^{n}{a_{ij}A_{ij}}$$

For some $i \in [1,n]$. $$

Properties of determinants

• $\big|A^{T}\big|=|A|$
• Every element of a row or column of a matrix is $0$ then the value of its determinant is $0$.
• If 2 columns or 2 rows of a matrix are identical then its determinant is $0$.
• If A and B are two square matrices then $\lvert{AB}\rvert=\lvert{A}\rvert\lvert{B}\rvert$.
• The value of the determinant of a matrix remains unchanged if a scalar multiple of a row or column is added to any other row or column.
• If a matrix $B$ is obtained from a square matrix $A$ by an interchange of two columns or rows: $\lvert{B}\rvert=−\lvert{A}\rvert$.
• If every entry in any row or column is multiplied by $k$, then the whole determinant is multiplied by $k$.

Composition

$$\begin{vmatrix} a & b & c_1 + c_2 \\ d & e & f_1 + f_2 \\ g & h & i_1 + i_2 \end{vmatrix} = \begin{vmatrix} a & b & c_1 \\ d & e & f_1 \\ g & h & i_1 \end{vmatrix} + \begin{vmatrix} a & b & c_2 \\ d & e & f_2 \\ g & h & i_2 \end{vmatrix}$$

In relation with eigenvalues

For a $n\times n$ matrix A with $n$ number of eigenvalues:

$$|A|=\prod_{i=1}^{n}{\lambda_i}$$