If $\mathrm{A}=\left(\begin{array}{ccc}1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5\end{array}\right), \mathrm{B}=\left(\begin{array}{c}-1 \\ -2 \\ 4\end{array}\right)$ and $[\mathrm{x} \mathrm{y} \mathrm{z}] \mathrm{A}^{\mathrm{T}}=\mathrm{B}^{\mathrm{T}}$, then $x+y+z=$

• A

  $4$
• B

  $-2$
• Correct answer
  C

  $6$
• D

  $3$

Solution

$\begin{aligned} & \Rightarrow\left[\begin{array}{ll}x y z\end{array}\right]\left[\begin{array}{ccc}1 & 2 & 3 \\ 5 & 4 & -1 \\ 3 & 0 & -5\end{array}\right]=\left[\begin{array}{lll}-1 & -24\end{array}\right] \\ & \Rightarrow \quad[x+5 y+3 z \quad 2 x+4 y \quad 3 x-y-5 z]=\left[\begin{array}{lll}-1 & -2 & 4\end{array}\right]\end{aligned}$ $\therefore \quad x+5 y+3 z=-1$ ...(i) $2 x+4 y=-2, \Rightarrow x+2 y=-1$ ...(ii) $3 x-y-5 z=4$ ...(iii) Solving eqs. (i), (ii) $\&$ (iii), we get $x=6, y=-7 / 2, z=7 / 2$ $\therefore x+y+z=6-\frac{7}{2}+\frac{7}{2}=6$

Hence, the correct answer is option: 'C'.