Question

If the matrix \[ M= \begin{bmatrix} x+5 & a & -4 \\ -2 & 0 & b \\ c & 6 & y+1 \end{bmatrix} \] is a skew-symmetric matrix, then the value of \[ ab+c^2-xy \] is:

• A. $-1$
• B. $-33$
• C. $-9$
• D. $0$

Hint:

In skew-symmetric matrices, first make all diagonal entries zero. It immediately gives equations for unknown variables.

Answer 1

The Correct Option is A

Concept: A square matrix $A$ is said to be skew-symmetric if: \[ A^T=-A \] Important properties of skew-symmetric matrices are:
• Every diagonal element is zero.
• Corresponding off-diagonal elements are negatives of each other. We use these properties to determine all unknown variables.

Step 1: Using the diagonal property.
In a skew-symmetric matrix: \[ a_{ii}=0 \] Therefore: \[ x+5=0 \] \[ x=-5 \] Also: \[ y+1=0 \] \[ y=-1 \]

Step 2: Comparing off-diagonal elements.
For skew-symmetric matrices: \[ a_{ij}=-a_{ji} \] Comparing entries: \[ a=-(-2) \] \[ a=2 \] Next, \[ -4=-c \] \[ c=4 \] Next, \[ b=-6 \]

Step 3: Substituting into the expression.
We need: \[ ab+c^2-xy \] Substitute values: \[ =(2)(-6)+(4)^2-(-5)(-1) \] \[ =-12+16-5 \] \[ =4-5 \] \[ =-1 \] Hence, \[ \boxed{-1} \]