Mathematics Exam  >  Mathematics Questions  >  Let M be the set of all invertible 5 × 5 matr... Start Learning for Free
Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then?
Most Upvoted Answer
Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1...
Introduction:

We are given a set M of all invertible 5 × 5 matrices with entries 0 and 1. We need to find the number of 1's and 0's in each matrix of M.

Explanation:

To find the number of 1's and 0's in each matrix of M, we need to understand the properties of invertible matrices and the possible combinations of 0's and 1's in such matrices.

Properties of Invertible Matrices:
- Invertible matrices have a non-zero determinant.
- The determinant of a matrix can be calculated using the cofactor expansion method or by using row operations to transform the matrix into echelon form.

Possible Combinations of 0's and 1's:
- Since the matrices in M are invertible, they must have a non-zero determinant. This implies that the matrices cannot have any row or column consisting entirely of 0's.
- Let's consider the number of 1's and 0's in each row and column of a matrix in M.
- If a row or column has all 1's, the determinant of the matrix will be 0. Therefore, each row and column must have at least one 0.
- Let's assume the number of 1's in a row is a and the number of 0's is b. Similarly, the number of 1's in a column is c and the number of 0's is d.
- Since each row and column must have at least one 0, we have a + b = 5 and c + d = 5.
- Also, the number of 1's and 0's in each row and column must add up to 5. Therefore, a + c = b + d = 5.

Calculating n0(M) and n1(M):
- From the above deductions, we can conclude that there are two possibilities for the number of 1's and 0's in each row and column: (1,4) or (2,3).
- For (1,4), there are 5 choices for the position of 0 in each row and column. Therefore, the total number of matrices with (1,4) combination is 5^5.
- For (2,3), there are 5 choices for the positions of 0's in each row and column. Therefore, the total number of matrices with (2,3) combination is 5^2 * 5^3.
- Hence, n0(M) = 5^5 + 5^2 * 5^3 and n1(M) = 5^5 + 5^2 * 5^3.

Summary:
- The number of 1's and 0's in each matrix of M can be determined by considering the properties of invertible matrices and the possible combinations of 0's and 1's in such matrices.
- From the analysis, we found that n0(M) = 5^5 + 5^2 * 5^3 and n1(M) = 5^5 + 5^2 * 5^3.
Explore Courses for Mathematics exam
Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then?
Question Description
Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then?.
Solutions for Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? in English & in Hindi are available as part of our courses for Mathematics. Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? defined & explained in the simplest way possible. Besides giving the explanation of Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then?, a detailed solution for Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? has been provided alongside types of Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? theory, EduRev gives you an ample number of questions to practice Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each and n0(M) denote the number of 1’s and 0’s in M, respectively. Then? tests, examples and also practice Mathematics tests.
Explore Courses for Mathematics exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev