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

Determinant

1. The symbol Image may be NSFW.
Clik here to view. $\left| \begin{array}{ll}{a_{1}} & {b_{1}} \\ {a_{2}} & {b_{2}}\end{array}\right|$ is called the determinant of order two. Its value is given by : D = a₁ b₂ − a₂ b₁

2. The symbol Image may be NSFW.
Clik here to view. $\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|$ s called the determinant of order three .
Its value can be found as:
Image may be NSFW.
Clik here to view. $\mathrm{D}=\mathrm{a}_{1} \left| \begin{array}{cc}{\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|-\mathrm{a}_{2} \left| \begin{array}{cc}{\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|+\mathrm{a}_{3} \left| \begin{array}{cc}{\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{b}_{2}} & {\mathrm{c}_{2}}\end{array}\right|$
OR
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Clik here to view. $\mathrm{D}=\mathrm{a}_{1} \left| \begin{array}{cc}{\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|-\mathrm{b}_{1} \left| \begin{array}{cc}{\mathrm{a}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{c}_{3}}\end{array}\right|+\mathrm{c}_{1} \left| \begin{array}{ll}{\mathrm{a}_{2}} & {\mathrm{b}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}}\end{array}\right|\ldots\ldots \text {and so on}$ .
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)
Image may be NSFW.
Clik here to view. $\text {are concurrent if}\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|=0$
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
Image may be NSFW.
Clik here to view. $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0=\left| \begin{array}{lll}{a} & {h} & {g} \\ {h} & {b} & {f} \\ {g} & {f} & {c}\end{array}\right|$
(iii) Area of a triangle whose vertices are (x_r, y_r) ; r = 1 , 2 , 3 is :
Image may be NSFW.
Clik here to view. $\mathrm{D}=\frac{1}{2} \left| \begin{array}{lll}{\mathrm{x}_{1}} & {\mathrm{y}_{1}} & {1} \\ {\mathrm{x}_{2}} & {\mathrm{y}_{2}} & {1} \\ {\mathrm{x}_{3}} & {\mathrm{y}_{3}} & {1}\end{array}\right|\text {If D = 0 then the three points are collinear.}$
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Clik here to view. $(iv) \text {Equation of a straight line passsing through}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \&\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right) \text { is } \left| \begin{array}{lll}{\mathrm{x}} & {\mathrm{y}} & {1} \\ {\mathrm{x}_{1}} & {\mathrm{y}_{1}} & {1} \\ {\mathrm{x}_{2}} & {\mathrm{y}_{2}} & {1}\end{array}\right|=0$

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,
Image may be NSFW.
Clik here to view. $\text {the minor of a 1 in (Key Concept 2) is}\left| \begin{array}{ll}{\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \& \text { the minor of } \mathrm{b}_{2} \text { is } \left| \begin{array}{ll}{\mathrm{a}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{3}} & {\mathrm{c}_{3}}\end{array}\right|$
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.
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Clik here to view. $\text {if D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|=\left| \begin{array}{ccc}{\mathrm{a}_{1}} & {\mathrm{a}_{2}} & {\mathrm{a}_{3}} \\ {\mathrm{b}_{1}} & {\mathrm{b}_{2}} & {\mathrm{b}_{3}} \\ {\mathrm{c}_{1}} & {\mathrm{c}_{2}} & {\mathrm{c}_{3}}\end{array}\right|=\mathrm{D}^{\prime} \mathrm{D} \& \mathrm{D}^{\prime}\text { are transpose of each other.}$
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.
Image may be NSFW.
Clik here to view. $\text {Let D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \quad \& \quad \mathrm{D}^{\prime}=\left| \begin{array}{lll}{\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {a_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|$
Then D′ = − D.
Property 3: If a determinant has any two rows (or columns) identical , then its value is zero . e. g.
Image may be NSFW.
Clik here to view. $\text {Let D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|$
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.
Image may be NSFW.
Clik here to view. $\text {Let D}=\left| \begin{array}{ccc}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \text { and } \mathrm{D}^{\prime}=\left| \begin{array}{lll}{\mathrm{Ka}_{1}} & {\mathrm{Kb}_{1}} & {\mathrm{Kc}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|$
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.
Image may be NSFW.
Clik here to view. $\left| \begin{array}{ccc}{a_{1}+x} & {b_{1}+y} & {c_{1}+z} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|=\left| \begin{array}{ccc}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|+\left| \begin{array}{ccc}{x} & {y} & {z} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|$
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.
Image may be NSFW.
Clik here to view. $\text {Let D}=\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right| \text { and } D^{\prime}=\left| \begin{array}{ccc}{a_{1}+m a_{2}} & {b_{1}+m b_{2}} & {c_{1}+m c_{2}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}+n a_{1}} & {b_{3}+n b_{1}} & {c_{3}+n c_{1}}\end{array}\right|$
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:
Image may be NSFW.
Clik here to view. $(i)\left| \begin{array}{ll}{a_{1}} & {b_{1}} \\ {a_{2}} & {b_{2}}\end{array}\right| \times \left| \begin{array}{ll}{1_{1}} & {m_{1}} \\ {l_{2}} & {m_{2}}\end{array}\right|=\left| \begin{array}{ll}{a_{1} l_{1}+b_{1} l_{2}} & {a_{1} m_{1}+b_{1} m_{2}} \\ {a_{2} l_{1}+b_{2} l_{2}} & {a_{2} m_{1}+b_{2} m_{2}}\end{array}\right|$
Similarly two determinants of order three are multiplied.
Image may be NSFW.
Clik here to view. $\text {If D}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right| \neq 0 \text { then }, \mathrm{D}^{2}=\left| \begin{array}{lll}{\mathrm{A}_{1}} & {\mathrm{B}_{1}} & {\mathrm{C}_{1}} \\ {\mathrm{A}_{2}} & {\mathrm{B}_{2}} & {\mathrm{C}_{2}} \\ {\mathrm{A}_{3}} & {\mathrm{B}_{3}} & {\mathrm{C}_{3}}\end{array}\right|$
where A_i, B_i, C_i are cofactors
Proof: Consider
Image may be NSFW.
Clik here to view. $\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right| \times \left| \begin{array}{ccc}{A_{1}} & {A_{2}} & {A_{3}} \\ {B_{1}} & {B_{2}} & {B_{3}} \\ {C_{1}} & {C_{2}} & {C_{3}}\end{array}\right|=\left| \begin{array}{ccc}{D} & {0} & {0} \\ {0} & {D} & {0} \\ {0} & {0} & {D}\end{array}\right|$
Note : a₁A₂ + b₁B₂ + c₁C₂ = 0 etc. therefore,
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Clik here to view. $\mathbf{D} \times \left| \begin{array}{lll}{A_{1}} & {A_{2}} & {A_{3}} \\ {B_{1}} & {B_{2}} & {B_{3}} \\ {C_{1}} & {C_{2}} & {C_{3}}\end{array}\right|=D^{3}\Rightarrow \left| \begin{array}{lll}{\mathrm{A}_{1}} & {\mathrm{A}_{2}} & {\mathrm{A}_{3}} \\ {\mathrm{B}_{1}} & {\mathrm{B}_{2}} & {\mathrm{B}_{3}} \\ {\mathrm{C}_{1}} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}}\end{array}\right|=\mathrm{D}^{2}\text {OR}\left| \begin{array}{ccc}{\mathrm{A}_{1}} & {\mathrm{B}_{1}} & {\mathrm{C}_{1}} \\ {\mathrm{A}_{2}} & {\mathrm{B}_{2}} & {\mathrm{C}_{2}} \\ {\mathrm{CA}_{3}} & {\mathrm{B}_{3}} & {\mathrm{C}_{3}}\end{array}\right|=\mathrm{D}^{2}$

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:
Image may be NSFW.
Clik here to view. $\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{b}_{2}} \neq \frac{\mathrm{c}_{1}}{\mathrm{c}_{2}} \Rightarrow\text { Given equations are inconsistent}$
&
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Clik here to view. $\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{b}_{2}}=\frac{\mathrm{c}_{1}}{\mathrm{c}_{2}} \Rightarrow \text {Given equations are dependent}$

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,
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Clik here to view. $\mathrm{x}=\frac{\mathrm{D}_{1}}{\mathrm{D}} \quad, \quad \mathrm{Y}=\frac{\mathrm{D}_{2}}{\mathrm{D}} \quad, \quad \mathrm{Z}=\frac{\mathrm{D}_{3}}{\mathrm{D}}$
Where
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Clik here to view. $D=\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|;$ Image may be NSFW.
Clik here to view. $D_{1}=\left| \begin{array}{lll}{\mathrm{d}_{1}} & {\mathrm{b}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{d}_{2}} & {\mathrm{b}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{d}_{3}} & {\mathrm{b}_{3}} & {\mathrm{c}_{3}}\end{array}\right|;$ Image may be NSFW.
Clik here to view. $\mathrm{D}_{2}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{d}_{1}} & {\mathrm{c}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{d}_{2}} & {\mathrm{c}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{d}_{3}} & {\mathrm{c}_{3}}\end{array}\right|$ Image may be NSFW.
Clik here to view. $\& \mathrm{D}_{3}=\left| \begin{array}{lll}{\mathrm{a}_{1}} & {\mathrm{b}_{1}} & {\mathrm{d}_{1}} \\ {\mathrm{a}_{2}} & {\mathrm{b}_{2}} & {\mathrm{d}_{2}} \\ {\mathrm{a}_{3}} & {\mathrm{b}_{3}} & {\mathrm{d}_{3}}\end{array}\right|$
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
Image may be NSFW.
Clik here to view. $\left.\begin{array}{l}{a_{1} x+b_{1} y+c_{1} z=d_{1}} \\ {a_{2} x+b_{2} y+c_{2} z=d_{2}} \\ {a_{3} x+b_{3} y+c_{3} z=d_{3}}\end{array}\right\}$
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
Image may be NSFW.
Clik here to view. $\left| \begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|=0.$
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.

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