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 R1  R2  and  R3       respectively

If we add R1    to   R2  and   R1   to  R3   then we get zero in 1st column, so  Operating  R1  → R1   + R2   and    R3  → 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 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 - R2  and  R3 ➡️ R1 - R3

Expanding along  C1

Δ = (xy2 )( xz3 ) - (xy3 )(xz2 )

Δ = (xy )(xy)(xz )( xz2  + xz ) - (xy )( xy2  + xy ) (x z )(x +z )

Δ = (xy )(xz )[(xy)( xz2  + xz ) -( xy2  + xy ) (x +z )]
Cancelling the same colour terms in the previous line ,then we have
Δ = (xy )(xz )[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 = (xy )(xz )[(yz2 - y2z) +( xz2  - y2x)]
Δ = (xy )(xz )[yz(z - y) + x(z2  - y2)]
Δ = (xy )(xz )[yz(z - y) + x(z - y)(zy)]
Δ = (xy )(xz )(z - y)[yz + x(zy)]
Δ = (xy )(xz )(z - y)[yz + xz+ xy]
Taking -1 common from (xz )(z - y) in previous line ,
Δ = (xy )(yz )(z - x)[yz + xz+ xy]

Hence the   proof

Final Words

Thanks for investing your precious time to read this post containing  Solving Determinants using elementary transformations,Short trick of elementary transformation  , elementary row transformation questions. If you liked it then share it with your near and dear ones to benefit them. we shall meet in next post with another beneficial article till then bye ,take care.......

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Share:

1 comment:

1. 