Determinant Definition, Properties, Formulas, Rules, Verification, Examples

Determinant

1. The symbol \(\left| \begin{array}{ll}{a_{1}} & {b_{1}} \\ {a_{2}} & {b_{2}}\end{array}\right|\) is called the determinant of order two. Its value is given by : D = a₁  b₂   − a₂  b₁

2. The symbol \(\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|\)  s called the determinant of order three .
Its value can be found as:
\(\mathrm{D}=\mathrm{a}_{1} \left| \begin{array}{cc}{\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|-\mathrm{a}_{2} \left| \begin{array}{cc}{\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|+\mathrm{a}_{3} \left| \begin{array}{cc}{\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{b}_{2}} & {\mathrm{c}_{2}}\end{array}\right|\)
OR
\(\mathrm{D}=\mathrm{a}_{1} \left| \begin{array}{cc}{\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|-\mathrm{b}_{1} \left| \begin{array}{cc}{\mathrm{a}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{c}_{3}}\end{array}\right|+\mathrm{c}_{1} \left| \begin{array}{ll}{\mathrm{a}_{2}} & {\mathrm{b}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}}\end{array}\right|\ldots\ldots \text {and so on}\).
In this manner we can expand a determinant in 6 ways using elements of ; R₁ , R₂ , R₃ or C₁ , C₂ , C₃.

3. Following examples of short hand writing large expressions are :
(i) The lines:
a₁x + b₁y + c₁ = 0…….. (1 )
a₂x + b₂y + c₂ = 0…….. (2)
a₃x + b₃y + c₃ = 0…….. (3)
\(\text {are concurrent if}\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|=0\)
Condition for the consistency of three simultaneous linear equations in 2 variables.
(ii) ax² + 2 hxy + by² + 2 gx + 2 fy + c = 0 represents a pair of straight lines if
\(a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0=\left| \begin{array}{lll}{a} & {h} & {g} \\ {h} & {b} & {f} \\ {g} & {f} & {c}\end{array}\right|\)
(iii) Area of a triangle whose vertices are (x_r, y_r) ; r = 1 , 2 , 3 is :
\(\mathrm{D}=\frac{1}{2} \left| \begin{array}{lll}{\mathrm{x}_{1}} & {\mathrm{y}_{1}} & {1} \\ {\mathrm{x}_{2}} & {\mathrm{y}_{2}} & {1} \\ {\mathrm{x}_{3}} & {\mathrm{y}_{3}} & {1}\end{array}\right|\text {If D = 0 then the three points are collinear.}\)
\((iv) \text {Equation of a straight line passsing through}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \&\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right) \text { is } \left| \begin{array}{lll}{\mathrm{x}} & {\mathrm{y}} & {1} \\ {\mathrm{x}_{1}} & {\mathrm{y}_{1}} & {1} \\ {\mathrm{x}_{2}} & {\mathrm{y}_{2}} & {1}\end{array}\right|=0\)

4. Minors: The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands For example,
\(\text {the minor of a 1 in (Key Concept 2) is}\left| \begin{array}{ll}{\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \& \text { the minor of } \mathrm{b}_{2} \text { is } \left| \begin{array}{ll}{\mathrm{a}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{3}} & {\mathrm{c}_{3}}\end{array}\right|\)
Hence a determinant of order two will have “4 minors” & a determinant of order three will have “9 minors” .

5. Cofactor: If M_ij represents the minor of some typical element then the cofactor is defined as: C_ij = (−1)^i+j . M_ij ; Where i & j denotes the row & column in which the particular element lies. Note that the value of a determinant of order three in terms of ‘Minor’ & ‘Cofactor’ can be written as : D = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃ OR D = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ & so on …….

6. Properties Of Determinants:

• Property 1: The value of a determinant remains unaltered , if the rows & columns are inter changed . e.g.
  \(\text {if D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|=\left| \begin{array}{ccc}{\mathrm{a}_{1}} & {\mathrm{a}_{2}} & {\mathrm{a}_{3}} \\ {\mathrm{b}_{1}} & {\mathrm{b}_{2}} & {\mathrm{b}_{3}} \\ {\mathrm{c}_{1}} & {\mathrm{c}_{2}} & {\mathrm{c}_{3}}\end{array}\right|=\mathrm{D}^{\prime} \mathrm{D} \& \mathrm{D}^{\prime}\text { are transpose of each other.}\)
  If D′ = − D then it is Skew Symmetric determinant but D′ = D ⇒ 2 D = 0 ⇒ D = 0 ⇒ Skew symmetric determinant of third order has the value zero.
• Property 2: If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g.
  \(\text {Let D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \quad \& \quad \mathrm{D}^{\prime}=\left| \begin{array}{lll}{\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {a_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|\)
  Then D′ = − D.
• Property 3: If a determinant has any two rows (or columns) identical , then its value is zero . e. g.
  \(\text {Let D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|\)
  then it can be verified that  D=0
• Property 4: If all the elements of any row (or column) be multiplied by the same number , then the determinant is multiplied by that number.
  e.g.
  \(\text {Let D}=\left| \begin{array}{ccc}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \text { and } \mathrm{D}^{\prime}=\left| \begin{array}{lll}{\mathrm{Ka}_{1}} & {\mathrm{Kb}_{1}} & {\mathrm{Kc}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|\)
  Then D′ = KD
• Property 5: If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants . e.g.
  \(\left| \begin{array}{ccc}{a_{1}+x} & {b_{1}+y} & {c_{1}+z} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|=\left| \begin{array}{ccc}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|+\left| \begin{array}{ccc}{x} & {y} & {z} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|\)
• Property 6: The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column) .e.g.
  \(\text {Let D}=\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right| \text { and } D^{\prime}=\left| \begin{array}{ccc}{a_{1}+m a_{2}} & {b_{1}+m b_{2}} & {c_{1}+m c_{2}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}+n a_{1}} & {b_{3}+n b_{1}} & {c_{3}+n c_{1}}\end{array}\right|\)
  Then D′ = D.
  Note: that while applying this property Atleast One Row (Or Column)  must remain unchanged.
• Property 7: If by putting x = a the value of a determinant vanishes then (x − a) is a factor of the determinant.

7.Multiplication Of Two Determinants:
\((i)\left| \begin{array}{ll}{a_{1}} & {b_{1}} \\ {a_{2}} & {b_{2}}\end{array}\right| \times \left| \begin{array}{ll}{1_{1}} & {m_{1}} \\ {l_{2}} & {m_{2}}\end{array}\right|=\left| \begin{array}{ll}{a_{1} l_{1}+b_{1} l_{2}} & {a_{1} m_{1}+b_{1} m_{2}} \\ {a_{2} l_{1}+b_{2} l_{2}} & {a_{2} m_{1}+b_{2} m_{2}}\end{array}\right|\)
Similarly two determinants of order three are multiplied.
\(\text {If D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \neq 0 \text { then }, \mathrm{D}^{2}=\left| \begin{array}{lll}{\mathrm{A}_{1}} & {\mathrm{B}_{1}} & {\mathrm{C}_{1}} \\ {\mathrm{A}_{2}} & {\mathrm{B}_{2}} & {\mathrm{C}_{2}} \\ {\mathrm{A}_{3}} & {\mathrm{B}_{3}} & {\mathrm{C}_{3}}\end{array}\right|\)
where A_i, B_i, C_i are cofactors
Proof: Consider
\(\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right| \times \left| \begin{array}{ccc}{A_{1}} & {A_{2}} & {A_{3}} \\ {B_{1}} & {B_{2}} & {B_{3}} \\ {C_{1}} & {C_{2}} & {C_{3}}\end{array}\right|=\left| \begin{array}{ccc}{D} & {0} & {0} \\ {0} & {D} & {0} \\ {0} & {0} & {D}\end{array}\right|\)
Note : a₁A₂ + b₁B₂ + c₁C₂ = 0 etc. therefore,
\(\mathbf{D} \times \left| \begin{array}{lll}{A_{1}} & {A_{2}} & {A_{3}} \\ {B_{1}} & {B_{2}} & {B_{3}} \\ {C_{1}} & {C_{2}} & {C_{3}}\end{array}\right|=D^{3}\Rightarrow \left| \begin{array}{lll}{\mathrm{A}_{1}} & {\mathrm{A}_{2}} & {\mathrm{A}_{3}} \\ {\mathrm{B}_{1}} & {\mathrm{B}_{2}} & {\mathrm{B}_{3}} \\ {\mathrm{C}_{1}} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}}\end{array}\right|=\mathrm{D}^{2}\text {OR}\left| \begin{array}{ccc}{\mathrm{A}_{1}} & {\mathrm{B}_{1}} & {\mathrm{C}_{1}} \\ {\mathrm{A}_{2}} & {\mathrm{B}_{2}} & {\mathrm{C}_{2}} \\ {\mathrm{CA}_{3}} & {\mathrm{B}_{3}} & {\mathrm{C}_{3}}\end{array}\right|=\mathrm{D}^{2}\)

8. System Of Linear Equation (In Two Variables):
(i) Consistent Equations: Definite & unique solution. [ intersecting lines ]
(ii) Inconsistent Equation: No solution. [ Parallel line ]
(iii) Dependent equation: Infinite solutions. [ Identical lines ]
Let a₁x + b₁y + c₁ = 0 & a₂x + b₂y + c₂ = 0 then:
\(\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{b}_{2}} \neq \frac{\mathrm{c}_{1}}{\mathrm{c}_{2}} \Rightarrow\text { Given equations are inconsistent}\)
&
\(\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{b}_{2}}=\frac{\mathrm{c}_{1}}{\mathrm{c}_{2}} \Rightarrow \text {Given equations are dependent}\)

9. Cramer’ S Rule :[ Simultaneous Equations Involving Three Unknowns ]
Let ,a₁x + b₁y + c₁z = d₁ …(I) ; a₂x + b₂y + c₂z = d₂ …(II) ; a₃x + b₃y + c₃z = d₃ …(III)
Then,
\(\mathrm{x}=\frac{\mathrm{D}_{1}}{\mathrm{D}} \quad, \quad \mathrm{Y}=\frac{\mathrm{D}_{2}}{\mathrm{D}} \quad, \quad \mathrm{Z}=\frac{\mathrm{D}_{3}}{\mathrm{D}}\)
Where
\(D=\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|;\)\(D_{1}=\left| \begin{array}{lll}{\mathrm{d}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{d}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{d}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|;\)\(\mathrm{D}_{2}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{d}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{d}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{d}_{3}} & {\mathrm{c}_{3}}\end{array}\right|\)\(\& \mathrm{D}_{3}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{d}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{d}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{d}_{3}}\end{array}\right|\)
Note: (a) If D ≠ 0 and alteast one of D₁ , D₂ , D₃ ≠ 0 , then the given system of equations are
consistent and have unique non trivial solution .
(b) If D ≠ 0 & D₁ = D₂ = D₃ = 0 , then the given system of equations are consistent and have trivial solution only
(c) If D = D₁ = D₂ = D₃ = 0 , then the given system of equations are consistentand have infinite solutions . In case
\(\left.\begin{array}{l}{a_{1} x+b_{1} y+c_{1} z=d_{1}} \\ {a_{2} x+b_{2} y+c_{2} z=d_{2}} \\ {a_{3} x+b_{3} y+c_{3} z=d_{3}}\end{array}\right\}\)
represents these parallel planes then also D = D₁ = D₂ = D₃ = 0 but the system is inconsistent.
(d) If D = 0 but atleast one of D₁ , D₂ , D₃ is not zero then the equations are inco ns istent and have no solution .

10. If x , y , z are not all zero , the condition for a₁x + b₁y + c₁z = 0 ; a₂x + b₂y + c₂z = 0 & a₃x + b₃y + c₃z = 0 to be consistent in x , y , z is that
\(\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|=0.\)
Remember that if a given system of linear equations have Only Zero Solution for all its variables then the given equations are said to have Trivial Solution.