$\text{If}\mathrm{ A}=\left[\begin{array}{ccc}0& -3& 2\\ a& b& 8\\ -2& c& 0\end{array}\right]\text{is skew-symmetric matrix, then}$

The correct option is: b = 0 and a + c = -5.

Step-by-Step Explanation:

A square matrix is said to be skew-symmetric if it is equal to the negative of its transpose. Mathematically, a matrix $A$ is skew-symmetric if:

$A^T=-A$

From this definition, we can derive two main properties for the elements $a_{ij}$ of a skew-symmetric matrix:
1. All the main diagonal elements must be zero ($a_{ii}=0$).
2. The non-diagonal elements satisfy the condition $a_{ji}=-a_{ij}$ for all $i\ne j$.

Let's apply these properties to the given matrix:

$A=\left[\begin{array}{ccc}0& -3& 2\\ a& b& 8\\ -2& c& 0\end{array}\right]$

Step 1: Determine the value of b using the diagonal property
The diagonal elements of matrix $A$ are $0$, $b$, and $0$. For the matrix to be skew-symmetric, the diagonal element in the second row, second column must be zero:
$b=0$

Step 2: Determine the value of a and c using the non-diagonal property
Using the relation $a_{ji}=-a_{ij}$:
For the element $A_{21}$ (second row, first column) and $A_{12}$ (first row, second column):

$a=-(-3)$

$a=3$

For the element $A_{32}$ (third row, second column) and $A_{23}$ (second row, third column):

$c=-8$

Step 3: Calculate a + c
Now, substitute the values of $a$ and $c$ to find $a+c$:

$a+c=3+(-8)=-5$

Therefore, we have $b=0$ and $a+c=-5$.