Using properties of determinants, prove that $$\left|\begin{array}{ccc}y+z & x & y \\ z+x & z & x \\ x+y & y & z\end{array}\right|=(x+y+z)(x-z)^{2}.$$

CBSE Class 12 · 2007 · DeterminantsLong Answer

Using properties of determinants, prove that

y + z z + x x + y x z y y x z = (x + y + z) (x - z)^{2} .

Answer: Proved

Apply

C_{1} \to C_{1} + C_{2} + C_{3}

; the first column becomes

x + y + z

throughout (since

(y + z) + x + y = x + 2 y + z

... ) — more cleanly, note

C_{1} + C_{3}

reproduces

x + y + z

patterns; take

(x + y + z)

common from

C_{1}

= (x + y + z) 111 x z y y x z .

Now

R_{2} \to R_{2} - R_{1}

R_{3} \to R_{3} - R_{1}

and expand along

C_{1}

= (x + y + z) [(z - x) (z - y) - (x - y) (y - x)] = (x + y + z) (x - z)^{2} .

Proved.

CBSE Class 12 Board · 2007 · Subjective