Math Problem Statement
Find the rank of the matrix 1 -3 4 -1 9 -2 6 -6 -1 -10 -3 9 -6 -6 -3 3 -3 4 9 0
Solution
We can find the rank of a matrix by performing row reduction (Gaussian elimination) to bring it to row echelon form, then counting the number of non-zero rows. The given matrix is:
1 & -3 & 4 & -1 & 9 \\ -2 & 6 & -6 & -1 & -10 \\ -3 & 9 & -6 & -6 & -3 \\ 3 & -3 & 4 & 9 & 0 \end{bmatrix}$$ Let's perform Gaussian elimination to find the rank. 1. We start by using the first row to eliminate elements in the first column of the other rows. 2. Then, we proceed similarly for the second column, and so on, reducing the matrix to row echelon form. I'll do the calculations to simplify this matrix.The rank of the given matrix is 3. Would you like any further details on the steps for Gaussian elimination, or do you have any other questions? Here are 5 related questions to expand on this topic: 1. How is Gaussian elimination performed step by step? 2. What is the relationship between rank and the number of pivots in a matrix? 3. Can the rank of a matrix change if we transpose it? 4. How does the rank of a matrix relate to its determinant? 5. What does the rank tell us about the solutions to a system of linear equations? **Tip:** The rank of a matrix also indicates the maximum number of linearly independent row (or column) vectors in the matrix.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Gaussian Elimination
Rank of a Matrix
Formulas
Gaussian elimination
Row echelon form
Rank of a matrix
Theorems
Rank-Nullity Theorem
Theorem on Linearly Independent Vectors
Suitable Grade Level
Grades 10-12, College Level