Math Problem Statement

compute the rank

Compute the rank of the given system of linear equations.

Solution

To find the rank of the given system of linear equations, we first convert the equations into an augmented matrix and then determine its rank by row reducing it to its row echelon form.

The given system of equations is:

3x_1 + 4x_2 = 11 \\ -6x_1 + 2x_2 + 2x_3 + 2x_4 = 12 \\ 9x_1 + 4x_2 - x_3 - x_4 = 10 \\ -x_1 + 2x_2 - x_4 = -1 \end{cases}$$ The corresponding augmented matrix is: $$\left[ \begin{array}{cccc|c} 3 & 4 & 0 & 0 & 11 \\ -6 & 2 & 2 & 2 & 12 \\ 9 & 4 & -1 & -1 & 10 \\ -1 & 2 & 0 & -1 & -1 \\ \end{array} \right]$$ Let's perform row reduction to determine the rank of this matrix. I will now compute the row echelon form.The rank of the given augmented matrix is 4, indicating that the system has full rank. This means all four rows are linearly independent. If you need further details or have any questions, feel free to ask. Here are five related questions to expand your understanding: 1. What does it mean if a system of linear equations has full rank? 2. How can the rank of a matrix help determine if a linear system has a unique solution? 3. What is the difference between row echelon form and reduced row echelon form? 4. How would the rank change if one row were a linear combination of the others? 5. Can the rank of a matrix exceed the number of columns? **Tip:** To find the rank of a matrix quickly, look for the number of non-zero rows in its row echelon form.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Rank
Systems of Linear Equations

Formulas

Matrix Row Echelon Form

Theorems

Rank-Nullity Theorem

Suitable Grade Level

College-level

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