Question

If $A=\left[\begin{array}{ccc}x& 1& 1\\ 1& x& 1\\ 1& 1& x\end{array}\right]$ and $B=\left[\begin{array}{cc}x& 1\\ 1& x\end{array}\right]$, then $\frac{dA}{dx}$=______.

• A. $3B+1$
• B. $3B$
• C. $-3B$
• D. $1-3B$

Answer 1

The correct option is B $3B$

Given that: $|A|=x(x^2-1)-(x-1)+(1-x)$ $=x^3-x-x+1+1-x$ $=x^3-3x+2$ ...(i)
Similarly the determinant of B is
$|B|=x^2-1$ ...(ii)
Now
$\frac{dA}{dx}=3x^2-3$ $=3(x^2-1)$ $=3|B|$