Question

If $ A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix} $ is symmetric and $ B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & 4 \\ -3 & r & s \end{bmatrix} $ is skew-symmetric, then find $ |A| + |B| - |AB| $

• A. \( xyz + pqr \)
• B. \( xyz + q + r \)
• C. \( \frac{xyz}{pq} \)
• D. \( xyz + pq + rs \)

Hint:

Use properties of symmetric and skew-symmetric matrices: \( a_{ij} = a_{ji} \), \( b_{ij} = -b_{ji} \), and \( b_{ii} = 0 \). These simplify filling unknowns quickly.

Answer 1

The Correct Option is B

To make matrix \( A \) symmetric: \[ a_{ij} = a_{ji} \Rightarrow x = 2,\ z = 5,\ y = 4 \Rightarrow A = \begin{bmatrix} -1 & 2 & -3 \\ 2 & 4 & 5 \\ 4 & 5 & -6 \end{bmatrix} \] So, \( x = 2,\ y = 4,\ z = 5 \Rightarrow xyz = 40 \) To make \( B \) skew-symmetric: \[ b_{ij} = -b_{ji},\ b_{ii} = 0 \Rightarrow p = -2,\ q = -q,\ s = 0,\ r = -4 \Rightarrow q = 0,\ s = 0,\ r = -4 \] Hence, \[ |A| + |B| - |AB| = xyz + q + r = 40 + 0 + (-4) = 36 \Rightarrow \text{Expression form: } xyz + q + r \]