JEE Mathematics 2023 Question with Solution

Given: A square matrix of order $5$ has entries from $\{0,1\}$. Each row sum is $1$ and each column sum is also $1$.

Find: The number of such matrices.

Such a matrix must have exactly one entry equal to $1$ in every row and exactly one entry equal to $1$ in every column. Therefore, it is a permutation matrix.

Choose the position of $1$ in the first row in $5$ ways. Then for the second row, only $4$ columns remain available. Similarly, for the third, fourth and fifth rows, the numbers of choices are $3,2,1$ respectively.

Therefore, the number of such matrices is $120$. The correct option is B.

Given: The matrix is of order $5$, entries are only $0$ or $1$, and every row sum and column sum equals $1$.

Find: How many such matrices exist.

A matrix satisfying these conditions has one and only one $1$ in each row, and because each column sum is also $1$, no two rows can place their $1$ in the same column.

So the matrix is completely determined by assigning to each row a distinct column containing the entry $1$. This is exactly a permutation of the $5$ columns.

The number of such assignments is the number of permutations of $5$ objects:

Hence, there are $120$ such matrices, and the correct option is B.