##
**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.

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

_{ }R

_{3 }

_{ }respectively

_{}If we add R

_{1 }

_{ to }

_{ }R

_{2}

_{ and }

_{ }R

_{1 }

_{ }to

_{ }R

_{3}

_{ then we get zero in 1st column, so }Operating R

_{1 → }R

_{1 }

_{ }+

_{ }R

_{2 }and

_{ }R

_{3 → }R

_{1 }

_{ }+

_{ }R

_{3}

_{ }

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

_{1 }

_{}△= (-a)×[(0)-(2c×2b)]

_{}△

_{= abc{-a(-4bc)}}

_{}△ = 4a

^{2}b

^{2}c

^{2}

Hence the proof

Watch this video for Understanding Elementary transformations

##
**PROBLEM**

Proof:- Put L H S of determinant to Î”

Operating R

_{1 }➡️xR_{1 }, R_{2 }➡️ yR_{2}and R_{3}➡️ zR_{3}
Taking common xyz from C

_{3}_{}

Operating R

_{2 }➡️ R_{1}- R_{2 and }R_{3 }➡️ R_{1}- R3Expanding along C

_{1}

Î” = (x

^{2 }- y^{2 })( x^{3 }- z^{3 }) - (x^{3 }- y^{3 })(x^{2 }- z^{2 })
Î” = (x- y

^{ })(x+ y)(x- z^{ })( x^{2 }+ z^{2 }+ xz ) - (x- y^{ })( x^{2 }+ y^{2 }+ xy ) (x^{ }- z^{ })(x^{ }+z^{ })
Î” = (x- y

^{ })(x- z^{ })**[**(x+ y)( x^{2 }+ z^{2 }+ xz ) -( x^{2 }+ y^{2 }+ xy ) (x^{ }+z^{ })**]**
Cancelling the same colour terms in the previous line ,then we have

Î” = (x- y

^{ })(x- z^{ })**[**xz^{2 }+ yz^{2 }- y^{2}x - y^{2}z^{ }**]**

Arranging terms in Squared Bracket in such a way that the term containing z

^{2 }must be at 1st and 3rd position and the term containing y^{2 }must be at 2nd and 4th position .
Î” = (x- y

^{ })(x- z^{ })**[(**yz^{2}- y^{2}z) +( xz^{2 }^{ }- y^{2}x)**]**
Î” = (x- y

^{ })(x- z^{ })**[**yz(z - y) + x(z^{2 }^{ }- y^{2})**]**
Î” = (x- y

^{ })(x- z^{ })**[**yz(z - y) + x(z - y)(z+ y)**]**
Î” = (x- y

^{ })(x- z^{ })(z - y)**[**yz + x(z+ y)**]**
Î” = (x- y

^{ })(x- z^{ })(z - y)**[**yz + xz+ xy**]**
Taking -1 common from (x- z

^{ })(z - y) in previous line ,
Î” = (x- y

^{ })(y- z^{ })(z - x)**[**yz + xz+ xy**]**

Hence the proof

## Final Words

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