Question

If $A=\left|\begin{array}{ccc}a+b& a+b+c& a+2+b+c\\ b+c& c+d+e& A+B+C+D\\ c+d& A+B+C& p+q+r+s\end{array}\right|$, then number of decomposition of $A$ is

• A. $12$
• B. $6$
• C. $8$
• D. $24$

Answer 1

The correct option is D $24$

Each element of first column is the sum of two elements.

$\therefore$ Number of decompostions of $C_1=2=l$ (say)

Similarly, number of decompostions of $C_2=3=m$ (say)

Number of decompostions of $C_3=4=n$ (say)

$\therefore$ Total number decomposition of $A$ is

$\therefore l\times m\times n=2\times 3\times 4=24$