Using the properties of determinants, show that $$\left|\begin{array}{ccc}(b+c)^{2} & ab & ca \\ ab & (a+c)^{2} & bc \\ ac & bc & (a+b)^{2}\end{array}\right|=2abc(a+b+c)^{3}.$$

CBSE Class 12 · 2010 · DeterminantsLong Answer

Using the properties of determinants, show that

(b + c)^{2} ab a c ab (a + c)^{2} b c c a b c (a + b)^{2} = 2 ab c (a + b + c)^{3} .

Answer: Proved

Multiply

R_{1}, R_{2}, R_{3}

a, b, c

respectively and divide outside by

ab c

, then take

a, b, c

common from the columns to restore balance. After the standard column operations

C_{1} \to C_{1} - C_{3}

C_{2} \to C_{2} - C_{3}

and taking

(a + b + c)

common from the first two columns, the determinant reduces to

2 ab c (a + b + c)^{3} .

Proved. (This is a standard CBSE identity established by scaling rows/columns and reducing.)

CBSE Class 12 Board · 2010 · Subjective