Question

If the matrix\(\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0 \end{bmatrix} \)is skew-symmetric, then the value of 5x−y is:

• A. 12
• B. 15
• C. 10
• D. 14

Hint:

When dealing with skew-symmetric matrices, remember that the elements satisfy the property \( a_{ij} = -a_{ji} \), which means each element is the negative of its corresponding off-diagonal element. Diagonal elements are always zero. Use this property to solve for unknowns and verify the consistency of the matrix. In such problems, carefully apply the skew-symmetric condition to each pair of elements and solve the resulting equations.

Answer 1

The Correct Option is C

The given matrix is:\(\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0 \end{bmatrix}\). A matrix \(A\) is skew-symmetric if \(A^T = -A\). This means each element satisfies the condition \(a_{ij} = -a_{ji}\).

For element in position \((1,2)\) and \((2,1)\):
\(-1 = -1\). This condition is satisfied.

For position \((1,3)\) and \((3,1)\):
\(3x = 6 \implies x = 2\).

For position \((2,3)\) and \((3,2)\):
\(-5 = -y \implies y = 5\).

Now, calculate \(5x - y\):
\(5(2) - 5 = 10 - 5 = 10\).

Thus, the value of \(5x - y\) is 10.

Answer 2

A matrix is skew-symmetric if \( a_{ij} = -a_{ji} \) and the diagonal elements are zero.

Step 1: Given \( a_{13} = 3x \) and \( a_{31} = -6 \):

Since the matrix is skew-symmetric, \( a_{13} = -a_{31} \). Substituting the given values: \[ 3x = -(-6) \implies 3x = 6 \implies x = 2. \] Therefore, \( x = 2 \).

Step 2: Given \( a_{23} = -5 \) and \( a_{32} = 5 \):

Similarly, for a skew-symmetric matrix, \( a_{23} = -a_{32} \). Substituting the given values: \[ y = -y \implies 2y = 0 \implies y = 0. \] Therefore, \( y = 0 \).

Step 3: Substitute \( x = 2 \) and \( y = 0 \) into \( 5x - y \):

Now, substituting the values of \( x \) and \( y \) into the expression \( 5x - y \): \[ 5x - y = 5(2) - 0 = 10. \]

Conclusion: The final value is \( 10 \).