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
- 225
- 120
- 125
- 150
The Correct Option is B
Approach Solution - 1
The correct answer is (B) : 120
Approach Solution -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:
- Place 1 in the first row in any of the 5 columns.
- Place 1 in the second row, avoiding the column already used by the first row.
- Place 1 in the third row, avoiding the columns already used by the first two rows.
- 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.
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.