Math Problem Statement
The rank of n × n matrix J, whose all entries above diagonal are 1, and other entries are 0 is n
n2
n2−n2
n2+n2 n-1
Solution
The matrix described in the question is an matrix where all the entries above the main diagonal are 1, and all other entries are 0. We can describe this matrix as:
0 & 1 & 1 & \cdots & 1 \\ 0 & 0 & 1 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 0 \\ \end{pmatrix}$$ To find the rank of this matrix, let's analyze the linear dependence of its rows or columns. - The first row contains all 1s from the second entry onward. - The second row has all 0s except for the second entry and beyond. - The third row has all 0s except for the third entry and beyond. The rows progressively contain fewer 1s as we move down, making them linearly dependent. Specifically, the rows from the second one onward are linear combinations of the rows above them. ### Conclusion: The rank of matrix $$J$$ is $$n - 1$$, since the first row is independent, and the subsequent rows span a subspace of dimension $$n-1$$. Thus, the correct answer is **$$n - 1$$**. --- Would you like more details on the reasoning behind this? Or do you have further questions about the rank of matrices? Here are some related questions to explore: 1. How do you determine the rank of a matrix using row reduction? 2. What is the definition of the rank of a matrix? 3. How can you find the rank of a matrix with all entries equal to 1? 4. How do row operations affect the rank of a matrix? 5. What is the rank of a diagonal matrix? **Tip**: For any matrix, the rank is the maximum number of linearly independent rows or columns.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Dependence
Matrix Theory
Formulas
Rank of matrix = maximum number of linearly independent rows or columns
Theorems
Rank of a matrix is the dimension of the column or row space
Suitable Grade Level
Grades 10-12