Math Problem Statement
Reduce the matrix to normal form
Solution
The image you provided contains a matrix with the instruction to reduce it to normal form, which likely refers to finding its row echelon form or reduced row echelon form.
The matrix appears as follows:
2 & -3 & 5 \\ 6 & -9 & 15 \\ -8 & -12 & 20 \end{pmatrix}$$ To reduce this matrix to normal form (possibly row echelon form), we will apply elementary row operations (swapping rows, multiplying a row by a scalar, and adding/subtracting rows). Let me perform the row reduction.The matrix $$A$$ provided has a rank of 2, meaning that its rows are not fully independent. This implies that one of the rows can be expressed as a linear combination of the others, leading to redundancy. However, I will proceed to reduce this matrix using Gaussian elimination. Let me calculate the row echelon form (or reduced row echelon form) of this matrix next.The reduced row echelon form (RREF) of the given matrix $$A$$ is: $$\text{RREF}(A) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -\frac{5}{3} \\ 0 & 0 & 0 \end{pmatrix}$$ This matrix is now in its normal form, where the leading 1s are positioned correctly, and each pivot column has zeros elsewhere. Would you like a more detailed explanation of the steps taken, or do you have further questions? Here are some additional questions for practice: 1. What are elementary row operations, and how do they work? 2. How is the rank of a matrix determined? 3. What are the differences between row echelon form and reduced row echelon form? 4. How does Gaussian elimination simplify solving linear systems? 5. Can a matrix always be reduced to row echelon form? **Tip:** When performing Gaussian elimination, start by choosing a pivot element (non-zero) and eliminate other entries in its column using elementary row operations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Row Reduction
Gaussian Elimination
Formulas
Elementary Row Operations
Theorems
Gaussian Elimination Theorem
Row Echelon Form
Reduced Row Echelon Form
Suitable Grade Level
College Level Mathematics