Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 3, 4, 5}. If the sum of all the entries of A is a prime number p, \(2 < p < 8\), then the number of such matrices A is ___________.
Correct Answer: 180
Solution and Explanation
\(∵ \) Sum of all entries of matrix A must be prime p such that \(2<p<8\) then sum of entries may be 3, 5 or 7.
If sum is 3 then possible entries are \((0, 0, 0, 3), (0, 0, 1, 2)\) or \((0, 1, 1, 1).\)
\(∴\) Total number of matrices \(= 4+4+12=20\)
If sum of 5 then possible entries are
\((0, 0, 0, 5), (0, 0, 1, 4), (0, 0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) \ and\ (1, 1, 1, 2).\)
\(∴\) Total number of matrices \(= 4+12+12+12+12+4=56\)
If sum is 7 then possible entries are
\((0, 0, 2, 5), (0, 0, 3, 4), (0, 1, 1, 5), (0, 3, 3, 1), (0, 2, 2, 3), (1, 1, 1, 4), (1, 2, 2, 2), (1, 1, 2, 3) \) and \((0, 1, 2, 4)\)
Total number of matrices with sum \(7=104\)
\(∴\) Total number of required matrices\(= 20+56+104=180\)
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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.