Math Fundamentals - Determinants

Second-Order Determinant

$\begin{vmatrix} a{11} & a{12} \newline a{21} & a{22} \newline \end{vmatrix}a{11}a{22}-a{12}a{21}$

Third-Order Determinant

$\begin{vmatrix}
a{11} & a{12} & a{13} \newline
a{21} & a{22} & a{23} \newline
a{31} & a{32} & a{33} \newline
\end{vmatrix}a{i1}A{i1}+a{i2}A{i2}+a{i3}A_{i3} \left( i=1,2,3 \right) $

or

$\begin{vmatrix}
a{11} & a{12} & a{13} \newline
a{21} & a{22} & a{23} \newline
a{31} & a{32} & a{33} \newline
\end{vmatrix}a{11}a{22}a{33}+a{12}a{23}a{31}+a{13}a{21}a{32} - a{11}a{23}a{32}-a{12}a{21}a{33}-a{13}a{22}a_{31}$

that is,

$\begin{vmatrix}
a{11} & a{12} & a{13} \newline
a{21} & a{22} & a{23} \newline
a{31} & a{32} & a{33} \newline
\end{vmatrix}a{11}$\begin{vmatrix} a_{22} & a_{23} \newline a_{32} & a_{33} \newline \end{vmatrix}-a_{12}$\begin{vmatrix}
a{21} & a{23} \newline
a{31} & a{33} \newline
\end{vmatrix}+a{13}$$\begin{vmatrix}
a{21} & a{22} \newline
a{31} & a_{32} \newline
\end{vmatrix}$

n-th Order Determinant

$
\begin{vmatrix}
a{11} & a{12} & \cdots & a{1n} \newline
a{21} & a{22} & \cdots & a{2n} \newline
\vdots & \vdots & & \vdots \newline
a{n1} & a{n2} & \cdots & a{nn} \newline
\end{vmatrix}
a{i1}A{i1}+a{i2}A{i2}+\cdots+a{in}A_{in} \left( i=1,2,\cdots ,n \right) $

In an n-th order determinant, the relationship between the minor and the cofactor of the element :

• In an n-th order determinant, the n-1 order determinant formed by the remaining elements after deleting the i-th row and j-th column where the element $a{ij}a{ij}M{ij}\left( -1 \right)^{i+j}M{ij}a{ij}a{ij}A_{ij}$, and the relationship between the two is

  $A{ij}=\left( -1 \right)^{i+j}M{ij}$

Properties of Determinants

• Property 1
  If the rows and columns of a determinant are interchanged, the value of the determinant stays the same

• Property 2
  If two rows (columns) of a determinant are interchanged, the value of the determinant changes sign

• Corollary 1
  If two rows (columns) of a determinant have identical corresponding elements, then the value of the determinant is zero

• Property 3
  Multiplying every element of a certain row (column) of a determinant by a number k is the same as multiplying the whole determinant by k

• Corollary 2
  If all the elements of a certain row (column) of a determinant share a common factor, then that common factor can be factored out in front of the determinant

• Corollary 3
  If two rows (columns) of a determinant have elements that are proportional to each other, then the value of the determinant is zero

• Property 4
  If the elements of a certain row (column) of a determinant are all the sum of two numbers, for example every element of the j-th column is the sum of two numbers:
  D=$\begin{vmatrix}
  a{11} & a{12} & \cdots & a{1j}+b{1j} & \cdots & a{1n} \newline
  a{21} & a{22} & \cdots & a{2j}+b{2j} & \cdots & a{2n} \newline
  \vdots & \vdots & & \vdots & & \vdots \newline
  a{n1} & a{n2} & \cdots & a{nj}+b{nj} & \cdots & a{nn} \newline
  \end{vmatrix}\begin{vmatrix}
  a{11} & a{12} & \cdots & a{1j} & \cdots & a{1n} \newline
  a{21} & a{22} & \cdots & a{2j} & \cdots & a{2n} \newline
  \vdots & \vdots & & \vdots & & \vdots \newline
  a{n1} & a{n2} & \cdots & a{nj} & \cdots & a{nn} \newline
  \end{vmatrix}\begin{vmatrix}
  a{11} & a{12} & \cdots & b{1j} & \cdots & a{1n} \newline
  a{21} & a{22} & \cdots & b{2j} & \cdots & a{2n} \newline
  \vdots & \vdots & & \vdots & & \vdots \newline
  a{n1} & a{n2} & \cdots & b{nj} & \cdots & a_{nn} \newline
  \end{vmatrix} \cdot $

  Note: +,
  =+
  =+++

• Corollary 4
  If every element of a certain row (column) of a determinant is written as the sum of m numbers (m being an integer greater than 2), then the determinant can be written as the sum of m determinants

• Property 5
  If all the elements of a certain row (column) of a determinant are multiplied by a number k and then added to the corresponding elements of another row (column), the value of the determinant stays the same

Formulas for Computing Determinants

• Upper triangular determinant:

  $\begin{vmatrix}
  & & & \ast & \newline
  a{11} & & & \newline
  & & a{22} & \newline
  & & & \ddots & \newline
  & & 0 & & a{nn} \newline
  \end{vmatrix} a{11}a{22} \cdots a{nn} $,
  where the elements in the “0” region are all zero, and likewise below.

• Lower triangular determinant:

  $\begin{vmatrix}
  & & & 0 & \newline
  a{11} & & & \newline
  & & a{22} & \newline
  & & & \ddots & \newline
  & & \ast & & a{nn} \newline
  \end{vmatrix} a{11}a{22} \cdots a{nn} \cdot$

• Diagonal determinant:

  $\begin{vmatrix}
  & & & 0 & \newline
  a{11} & & & \newline
  & & a{22} & \newline
  & & & \ddots & \newline
  & & 0 & & a{nn} \newline
  \end{vmatrix} a{11}a{22} \cdots a{nn} \cdot$

• Anti-diagonal determinant:

  $\begin{vmatrix}
  & 0 & & \newline
  & & & a{1n} \newline
  & & a{2,n-1} & \newline
  & \unicode{x22f0} & & \newline
  & a{n-1,2} & & \newline
  a{n1} & & 0 & \newline
  \end{vmatrix}$

• Block determinant:
  = ,

  =

Cramer’s Rule

If the coefficient determinant D of the system of linear equations $ \begin{cases} a{11}x_1 + a{12}x2 + \cdots + a{1n}xn = b_1 \newline
a{21}x1 + a{22}x2 + \cdots + a{2n}xn = b_2 \newline
\cdots\cdots \newline
a{n1}x1 + a{n2}x2 + \cdots + a{nn}xn = b_n \newline \end{cases}\begin{vmatrix}
a{11} & a{12} & \cdots & a{1n} \newline
a{21} & a{22} & \cdots & a{2n} \newline
\vdots & \vdots & & \vdots \newline
a{n1} & a{n2} & \cdots & a{nn} \newline
\end{vmatrix}\not=x_1=\frac{D_1}{D} x_2=\frac{D_2}{D} \cdots x_n=\frac{D_n}{D}\cdot$

Note: there are two cases in which Cramer’s Rule cannot be used:

1. The number of unknowns in the linear equations differs from the number of equations
2. The number of unknowns equals the number of equations, but the coefficient determinant equals zero