HOW TO SOLVE DETERMINANTS USING ELEMENTARY TRANSFORMATIONS

How to Prove Determinants using elementary transformations 

In this post we shall discuss Short trick of elementary transformation,Solving Determinants using elementary transformations,define elementary transformation, elementary transformation class 12, elementary row transformation questions.

By this method we have  to reduce maximum elements of specific Rows or column to zero, so that we can solve it easily

To solve the determinants using elementary transformations , Let us suppose L H S = △

As we can see that 'a' is common in 1st Row , 'b' is common in 2nd Row and 'c' is common in 3rd row ,
Therefore Taking a ,b ,c common from R_1  , R_2  , and  R₃       respectively

If we add R_1    to   R_2  and   R₁   to  R_{3   then we get zero in 1st column, so}  Operating  R_{1  →} R₁   + R₂   and    R_{3  →} R₁   + R₃  

_{As we have received maximum possible  zero in 1st column Therefore Expanding along} C₁  

△= (-a)×[(0)-(2c×2b)]
△ _{= abc{-a(-4bc)}}

△ = 4a²b²c² 

Hence the proof


Watch this video for Understanding Elementary transformations

PROBLEM

Proof:- Put L H S of determinant to Δ

Operating R₁ ➡️xR₁  , R₂ ➡️ yR₂ and R₃➡️ zR₃

Taking common xyz from C₃

Operating  R₂ ➡️ R₁ - R_2  and  R₃ ➡️ R₁ - R3

Expanding along  C₁ 

Δ = (x² - y² )( x³ - z³ ) - (x³ - y³ )(x² - z² )

Δ = (x- y )(x+ y)(x- z )( x² + z²  + xz ) - (x- y )( x² + y²  + xy ) (x - z )(x +z )

Δ = (x- y )(x- z )[(x+ y)( x² + z²  + xz ) -( x² + y²  + xy ) (x +z )]

Cancelling the same colour terms in the previous line ,then we have 

 Δ = (x- y )(x- z )[xz²   + yz² - y²x - y²z  ]

Arranging  terms in Squared Bracket  in such a way that the term containing z² must be at 1st and 3rd position and the term containing y² must be at 2nd and 4th position .

Δ = (x- y )(x- z )[(yz² - y²z) +( xz²  - y²x)]

Δ = (x- y )(x- z )[yz(z - y) + x(z²  - y²)]

Δ = (x- y )(x- z )[yz(z - y) + x(z - y)(z+ y)]

Δ = (x- y )(x- z )(z - y)[yz + x(z+ y)]

Δ = (x- y )(x- z )(z - y)[yz + xz+ xy]

Taking -1 common from (x- z )(z - y) in previous line ,

Δ = (x- y )(y- z )(z - x)[yz + xz+ xy]

Hence the   proof

Also Read   HOW TO FIND THE SLOPE OF LINE  AX + BY = C ,  SLOPE OF A  LINE

Final Words

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