Matrices & Determinants
Counting matrices with constraints
nta_pyq_2023_jan
Grade 12
Question:
The number of square matrices of order 5 with entries from the set $\{0, 1\}$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is
Step-by-Step Solution
Key Concept: Each row and column has exactly one 1 — these are permutation matrices, counted by $5!$
Each row must have exactly one 1 and each column must have exactly one 1. This is a permutation matrix of order 5. Number of such matrices $= 5! = 120$. Answer: (2)
Correct Answer: 120