If the matrix $\begin{bmatrix}-1&x-y&4\\2&0&5\\x+y&z&6\end{bmatrix}$ is symmetric, then $x + 3y+2z$ is equal to

16

The correct answer is Option (1) → 16 **

The matrix is symmetric:

$\begin{pmatrix} -1 & x-y & 4\\ 2 & 0 & 5\\ x+y & z & 6 \end{pmatrix}$

For symmetry: $a_{ij} = a_{ji}$.

Compare:

$a_{12} = a_{21}$ ⇒ $x - y = 2$ … (1)

$a_{13} = a_{31}$ ⇒ $4 = x + y$ … (2)

$a_{23} = a_{32}$ ⇒ $5 = z$ … (3)

Now, solve (1) and (2):

Add (1) and (2):

$(x - y) + (x + y) = 2 + 4$

$2x = 6$

$x = 3$

Substitute in (2):

$3 + y = 4$

$y = 1$

From (3):

$z = 5$

Now compute $x + 3y + 2z$:

$= 3 + 3(1) + 2(5)$ $= 3 + 3 + 10$ $= 16$

Final Answer: 16