By using properties of determinants, prove that $$\left|\begin{array}{lll}b+c & q+r & y+z \\ c+a & r+p & z+x \\ a+b & p+q & x+y\end{array}\right|=2\left|\begin{array}{lll}a & p & x \\ b & q & y \\ c & r & z\end{array}\right|.$$

CBSE Class 12 · 2012 · DeterminantsLong Answer

By using properties of determinants, prove that

b + c c + a a + b q + r r + p p + q y + z z + x x + y = 2 a b c p q r x y z .

Answer: Proved

Apply

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

; each entry of the new first row is twice a sum, e.g.

2 (a + b + c)

. Take

2

common:

= 2 a + b + c c + a a + b p + q + r r + p p + q x + y + z z + x x + y .

Now

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

gives row

(b, q, y)

(after suitable regrouping), and

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

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

reduce the rows to

(a, p, x), (b, q, y), (c, r, z)

= 2 a b c p q r x y z .

Proved.

CBSE Class 12 Board · 2012 · Subjective