Question:
The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _________.
The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _________.
Updated On: Jan 8, 2025
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Correct Answer: 282
Solution and Explanation
In a 3 × 3 order matrix there are 9 entries.
These nine entries are zero or one.
The sum of positive prime entries are 2, 3, 5 or 7.
Total possible matrices
=\(\frac{9!}{2!⋅7!}\)+\(\frac{9!}{3!⋅6!}\)+\(\frac{9!}{5!⋅4!}\)+\(\frac{9!}{2!⋅7!}\)
= 36 + 84 + 126 + 36
= 282
These nine entries are zero or one.
The sum of positive prime entries are 2, 3, 5 or 7.
Total possible matrices
=\(\frac{9!}{2!⋅7!}\)+\(\frac{9!}{3!⋅6!}\)+\(\frac{9!}{5!⋅4!}\)+\(\frac{9!}{2!⋅7!}\)
= 36 + 84 + 126 + 36
= 282
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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.