Matrices & Determinants
Determinant of sum of skew-symmetric matrices
MJMT_Full_Test_10
Grade 12
Question:
Let $A_k=[a_{ij}]$ be square matrix of order 3 with $a_{ij}=(i-j)^k$ for all $i,j\in\{1,2,3\}$. Determinant value of $|A_1+A_3+A_5+\cdots+A_{2023}|$ equals
$3^{4046}$
$3^{6069}$
$3^{6069}-3^{4046}$
none of these
Step-by-Step Solution
Key Concept: For odd $k$: $a_{ij}=(i-j)^k\Rightarrow a_{ji}=(j-i)^k=-(i-j)^k=-a_{ij}$. So $A_k$ is skew-symmetric for all odd $k$. Sum of skew-symmetric matrices is skew-symmetric. Determinant of odd-order skew-symmetric matrix $=0$.
$\det=0$ = none of these.
Correct Answer: 4