Permutations and Combinations
Permutations in Matrices
JEE Main 2021
Grade 11
Question:
The total number of $3 \times 3$ matrices $A$ with entries from $\{0, 1, 2, 3\}$ such that the sum of all diagonal entries of $AA^T$ is $4$ is:
(1) $135$
(2) $198$
(3) $162$
(4) $243$
Step-by-Step Solution
Key Concept: $tr(AA^T) = \sum_{i,j} a_{ij}^2 = 4$ (sum of squares of all 9 entries equals 4). Cases: one entry $= 2$, rest $0$: $\binom{9}{1} = 9$; four entries $= 1$, rest $0$: $\binom{9}{4} = 126$. Total $= 9 + 126 = 135$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)