Q21 of 26 Page 1

Using properties of determinants, prove that


OR


If and find k.

Multiplying R1, R2, R3 by z, x, y Respectively


⇒


Take common z, x, y from C1, C2 and C3


⇒


C1→C1 – C3 and C2→ C2 – C3


Taking common x + y + z from C1 and C2


⇒


R3→ R3 - (R1 + R2)


⇒


C1→ zC1 and C2 → xC3


⇒


C1→ C1 + C3 and C2⇒ C2 + C3


⇒


Taking z and x common from R1 and R2


⇒


Expansion along R3


= (x + y + z)2.(2xz((x + y)(z + y) - xz))


= (x + y + z)2.(2xz((xz + xy + yz + y2 - xz)


= (x + y + z)2.(2xz((xy + yz + y2)


= 2xyz(x + y + z)3


OR


⇒


⇒


⇒


⇒


⇒


∴ A3 - 6A2 + 7A + KI3 = 0


⇒


⇒ K = 2


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