Math Problem Statement
Beregn rangen af følgende matrix:
\[
\begin{bmatrix}
2 & 2 & 0 \\
-1 & 1 & 2 \\
0 & 2 & 2 \\
0 & 1 & 1 \\
\end{bmatrix}
\]
Solution
To find the rank of the given matrix, we need to determine the maximum number of linearly independent rows or columns in it. Here’s the matrix given:
2 & 2 & 0 \\ -1 & 1 & 2 \\ 0 & 2 & 2 \\ 0 & 1 & 1 \\ \end{bmatrix}$$ I'll perform row reduction on this matrix to bring it to its row echelon form, from which we can easily determine the rank by counting the non-zero rows. 1. Start with the first row and use it to eliminate elements below in the same column. 2. Continue with each subsequent row. Let’s proceed with the row reduction. ### Step-by-Step Solution 1. **Row Operations**: - Make the element in the first column of the second row zero by adding $$\frac{1}{2}$$ times the first row to the second row. - Similarly, use row operations to make other elements zero to simplify the matrix. After completing the row reduction, we'll count the number of non-zero rows to find the rank of the matrix. --- Let me compute this for you.The rank of the given matrix is $$2$$. This means there are two linearly independent rows (or columns) in the matrix. Would you like more details on the row reduction process or have any other questions? Here are some related questions for further exploration: 1. How do we determine the rank of a matrix by counting pivots? 2. What is the significance of the rank in terms of the solutions to a system of linear equations? 3. Can the rank of a matrix be greater than its number of rows or columns? 4. How does rank relate to the concept of linear independence? 5. What are other methods for finding the rank of a matrix, such as using determinants? **Tip**: The rank of a matrix also tells you about its nullity, as the two are related by the rank-nullity theorem.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Linear Algebra
Row Reduction
Rank of a Matrix
Formulas
Row operations for echelon form
Rank determination by non-zero rows
Theorems
Rank-Nullity Theorem
Fundamental Theorem of Linear Algebra
Suitable Grade Level
University level (Undergraduate)