Question

What is the value of the determinant $$\begin{vmatrix} 1 & bc & a\left( b+c \right) \\ 1 & ca & b\left( c+a \right) \\ 1 & ab & c\left( a+b \right) \end{vmatrix}$$?

A
0
B
abc
C
ab+bc+ca
D
abc(a+b+c)

Solution

The correct option is A $$0$$Now, in the matrix$$\begin{vmatrix} 1 & bc & ab+ac \\ 1 & ac & bc+ba \\ 1 & ab & ca+cb \end{vmatrix}$$$$C_3\rightarrow C_3+C_2$$$$\begin{vmatrix} 1 & bc & ab+ac+bc \\ 1 & ac & ab+ac+bc \\ 1 & ab & ab+ac+bc \end{vmatrix}$$    taking $$(ab+bc+ac)$$ common in $$C_3$$$$(ab+bc+ac)$$$$\begin{vmatrix} 1 & bc & 1 \\ 1 & ac & 1 \\ 1 & ab & 1 \end{vmatrix}$$Now two columns of the matrix are same so it's determinant will be 0.(a)Mathematics

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