Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade None
Question:
Consider a skew-symmetric matrix $A = \begin{bmatrix} 0 & b & k \\ b & 0 & c \\ -c & -c & 0 \end{bmatrix}$ such that $a, b$ and $c$ are selected from the set $S = \{0, 1, 2, 3, \ldots, 12\}$. If $|A|$ is divisible by 3, then the number of such possible matrices is
Step-by-Step Solution
Key Concept: For skew symmetric matrices, the determinant has special divisibility properties that constrain the parameter values.
Since $A$ is skew symmetric with $a = c = 0$, the matrix has the form with zeros on the diagonal and opposite entries off-diagonal. For a $3\times 3$ skew symmetric matrix, $|A| = b^2$ where $b$ is related to the off-diagonal entries. For $|A|$ to be divisible by 3, we need $b \equiv 0 \pmod{3}$. Testing the values $k \in \{0, 3, 6, 9, 12\}$, the constraint is satisfied for $k \in \{0, 3, 6, 9\}$.
Correct Answer: 1