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

• A. 225
• B. 120
• C. 125
• D. 150

Answer 1

The Correct Option is B

The correct answer is (B) : 120

In each row and each column exactly one is to be placed -
$\therefore$ No. of such materials $=5\times 4\times 3\times 2\times 1=120$
Step-1 : Select any 1 place for 1 's in row 1. Automatically some column will get filled with 0 's.
Step-2 : From next now select 1 place for 1's. Automatically some column will get filled with 0 's. $\Rightarrow$ Each time one less place will be available for putting 1's.
Repeat step-2 till last row.
Req. ways $=5\times 4\times 3\times 2\times 1=120$

Answer 2

Step 1: Understand the structure of a permutation matrix 

A permutation matrix is a square matrix where:

• Each row contains exactly one 1, and the rest are 0.
• Each column contains exactly one 1, and the rest are 0.

For a 5 × 5 permutation matrix, this means:

• Each row contains a 1 in one of the 5 columns.
• Each column contains a 1 in one of the 5 rows.

Example of a 5 × 5 permutation matrix:

\[ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} \] 

Step 2: Count the total number of permutation matrices

To construct a permutation matrix of order 5, we need to:

1. Place 1 in the first row in any of the 5 columns.
2. Place 1 in the second row, avoiding the column already used by the first row.
3. Place 1 in the third row, avoiding the columns already used by the first two rows.
4. Repeat this process for all rows.

This is equivalent to arranging 5 elements (columns) in all possible orders, which is the number of permutations of 5 objects.

The total number of permutations of \( n \) objects is given by:

\[ n! = n \times (n - 1) \times (n - 2) \times \dots \times 1. \] For \( n = 5 \): \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120. \]

Thus, there are 120 distinct 5 × 5 permutation matrices.

Step 3: Verify conditions

Let's verify that each matrix satisfies the given conditions:

• Each row contains exactly one 1 because we place 1 in one of the 5 columns for each row without repetition.
• Each column contains exactly one 1 because we use each column exactly once across all rows.

Therefore, all 120 matrices satisfy the given conditions.