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LHS |xx2yzyy2zxzz2xy|=1xyz |x·xx·x2x·yzy·yy·y2y·zxz·zz·z2z·xy| Multiplying R1, R2& R3 by x, y&z.=1xyz|x2x3xyzy2y3xyzz2z3xyz|.=xyzxyz|x2x31y2y31z2z31| Taking xyz common from c3.=|x2x31y2−x2y3−x31−1z2−x223−x31−1| 2→2−13→3−1 .= |x2x31(y−x)(y+x)(y−x)(y2+x2+xy)0(z−x)(z+x)(2−x)(z2+x2+2x)0|Expanding along c3 we get,= 1 |(y−x)(y+x)(y−x)(y2+x2+xy)(z−x)(z+x)(z−x)(z2+x2+zx)|= (y-x)(z-x). |y+xy2+x2+xyz+xz2+x2+zx| Taking (yx) & (zx) common from R1 and R2.= (y-x)(z-x). Taking (y-x) & (z-x) common from R1 and R2.= (y-x)(z-x)[(y + x)(z2 + x2 + zx) - (z + x)(y2 + x2 + xy)]= (y-x)(z-x)[yz2 + yx2 + xyz + xz2 + x3 + x2z-zy2-zx2-xyz-xy2-x3-x2y]= y-x)(z-x)[yz2-zy2 + xz2-xy2]= (y-x)(z-x)[yz (z-y) + x(z2-y2)]= y-x)(z-x)(z-y)[yz + x (z + y)] = (- 1)(x-y)(z-x)(- 1)(y-z)[yz+ 1 xz + xy]= (x-y)(y-z)(z-x)(xy + yz + xz). = R.H.S

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Ncert Solutions Maths class 12th Maths Class 12th Determinants

16. $| \begin{matrix} x & x^{2} & y z \\ y & y^{2} & z x \\ z & z^{2} & x y \end{matrix} | = (x - y) (y - z) (z - x) (x y + y z + z x)$

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Vishal Baghel | Contributor-Level 10

4 months ago

LHS $| \begin{matrix} x & x^{2} & y z \\ y & y^{2} & z x \\ z & z^{2} & x y \end{matrix} |$

$= \frac{1}{x y z}$ $| \begin{matrix} x \cdot x & x \cdot x^{2} & x \cdot y z \\ y \cdot y & y \cdot y^{2} & y \cdot z x \\ z \cdot z & z \cdot z^{2} & z \cdot x y \end{matrix} |$ Multiplying R1, R2& R3 by x, y&z.

$= \frac{1}{x y z} | \begin{matrix} x^{2} & x^{3} & x y z \\ y^{2} & y^{3} & x y z \\ z^{2} & z^{3} & x y z \end{matrix} | .$

$= \frac{x y z}{x y z} | \begin{matrix} x^{2} & x^{3} & 1 \\ y^{2} & y^{3} & 1 \\ z^{2} & z^{3} & 1 \end{matrix} |$ Taking xyz common from c3.

$= | \begin{matrix} x^{2} & x^{3} & 1 \\ y^{2} - x^{2} & y^{3} - x^{3} & 1 - 1 \\ z^{2} - x^{2} & 2^{3} - x^{3} & 1 - 1 \end{matrix} | \begin{matrix} _{2} \to_{2} -_{1} & _{3} \to_{3} -_{1} . \end{matrix}$

= $| \begin{matrix} x^{2} & x^{3} & 1 \\ (y - x) (y + x) & (y - x) (y^{2} + x^{2} + x y) & 0 \\ (z - x) (z + x) & (2 - x) (z^{2} + x^{2} + 2 x) & 0 \end{matrix} |$

Expanding along c3 we get,

= 1 $| \begin{matrix} (y - x) (y + x) & (y - x) (y^{2} + x^{2} + x y) \\ (z - x) (z + x) & (z - x) (z^{2} + x^{2} + z x) \end{matrix} |$

= (y-x)(z-x). $| \begin{matrix} y + x & y^{2} + x^{2} + x y \\ z + x & z^{2} + x^{2} + z x \end{matrix} |$ Taking (yx) & (zx) common from R₁ and R₂.

= (y-x)(z-x). Taking (y-x) & (z-x) common from R₁ and R₂.

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

= (y-x)(z-x)[yz² + yx² + xyz + xz² + x³ + x²z-zy²-zx²-xyz-xy²-x³-x²y]

= y-x)(z-x)[yz²-zy² + xz²-xy²]

= (y-x)(z-x)[yz (z-y) + x(z²-y²)]

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

= (- 1)(x-y)(z-x)(- 1)(y-z)[yz+ 1 xz + xy]

= (x-y)(y-z)(z-x)(xy + yz + xz). = R.H.S

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16. $| \begin{matrix} x & x^{2} & y z \\ y & y^{2} & z x \\ z & z^{2} & x y \end{matrix} | = (x - y) (y - z) (z - x) (x y + y z + z x)$

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16. |xx2yzyy2zxzz2xy|=(x−y)(y−z)(z−x)(xy+yz+zx)

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16. $| \begin{matrix} x & x^{2} & y z \\ y & y^{2} & z x \\ z & z^{2} & x y \end{matrix} | = (x - y) (y - z) (z - x) (x y + y z + z x)$