Question

If the matrix \(A =\begin{bmatrix}x&-2 &-5y\\ 2&0& -9 \\10& 3z &0\end{bmatrix}\)is skew-symmetric, then the value of \((2x-3y+4z)\) is:

• A. 5
• B. 6
• C. 0
• D. -2

Answer 1

The Correct Option is B

To solve this problem, we need to use the properties of a skew-symmetric matrix. A matrix \(A\) is skew-symmetric if \(A = -A^T\), where \(A^T\) is the transpose of \(A\). Additionally, the diagonal elements of a skew-symmetric matrix must be zero.

Given matrix \(A = \begin{bmatrix}x & -2 & -5y \\ 2 & 0 & -9 \\ 10 & 3z & 0\end{bmatrix}\), we can apply skew-symmetric properties as follows:

1. The elements must satisfy: \(a_{ij} = -a_{ji}\).
2. Diagonal elements must be zero (i.e., \(x = 0\) and the elements at (2,2), (3,3)).

Evaluating the conditions:

• Element (1,1): \(x = 0\) (since diagonal elements must be zero).
• Elements (1,2) and (2,1): \(-2 = -2\) (condition satisfied).
• Elements (1,3) and (3,1): \(-5y = -10\)
  \(5y = 10 \Rightarrow y = 2\).
• Elements (2,3) and (3,2): \(-9 = -(3z)\)
  \(3z = 9 \Rightarrow z = 3\).

Now we compute \(2x - 3y + 4z\) with \(x = 0\), \(y = 2\), and \(z = 3\):

\(2x - 3y + 4z = 2(0) - 3(2) + 4(3)\)

\(= 0 - 6 + 12\)

\(= 6\)

Therefore, the value of \(2x - 3y + 4z\) is 6.