Math Problem Statement
ispisi samo esalonsku i reduciranu esalonsku formu za svaki
Svedite matrice sustava na esalonsku formu (EF) i reduciranu esalonsku formu (REF):
Solution
Let's go through each matrix in the image, reducing them to their row echelon form (EF) and reduced row echelon form (REF). Here are the results for each part:
a) Given matrix:
1 & 2 & 3 & 4 \\ 2 & 3 & 8 & 4 \\ 3 & 2 & 17 & 1 \end{pmatrix}$$ - Row Echelon Form (EF): $$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & -1 & 2 & -4 \\ 0 & 0 & 9 & -19 \end{pmatrix}$$ - Reduced Row Echelon Form (REF): $$\begin{pmatrix} 1 & 0 & 0 & 17/9 \\ 0 & 1 & 0 & -29/9 \\ 0 & 0 & 1 & -19/9 \end{pmatrix}$$ --- ### b) Given matrix: $$\begin{pmatrix} 1 & -2 & 1 & 1 \\ 2 & 1 & -1 & 2 \\ 0 & 5 & -3 & 0 \end{pmatrix}$$ - Row Echelon Form (EF): $$\begin{pmatrix} 1 & -2 & 1 & 1 \\ 0 & 5 & -3 & 0 \\ 0 & 0 & 1 & 2 \end{pmatrix}$$ - Reduced Row Echelon Form (REF): $$\begin{pmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 6/5 \\ 0 & 0 & 1 & 2 \end{pmatrix}$$ --- ### c) Given matrix: $$\begin{pmatrix} 2 & 3 & -1 & 0 \\ 1 & -3 & 1 & 0 \\ 0 & -3 & 1 & 3 \end{pmatrix}$$ - Row Echelon Form (EF): $$\begin{pmatrix} 2 & 3 & -1 & 0 \\ 0 & -9 & 2 & 0 \\ 0 & 0 & 1 & 3 \end{pmatrix}$$ - Reduced Row Echelon Form (REF): $$\begin{pmatrix} 1 & 0 & 0 & -1.5 \\ 0 & 1 & 0 & -2/3 \\ 0 & 0 & 1 & 3 \end{pmatrix}$$ --- ### d) Given matrix: $$\begin{pmatrix} 4 & -1 & 1 & 11 \\ 2 & 3 & -1 & 3 \\ 1 & -2 & 1 & 4 \end{pmatrix}$$ - Row Echelon Form (EF): $$\begin{pmatrix} 4 & -1 & 1 & 11 \\ 0 & 4.5 & -1.5 & -2.5 \\ 0 & 0 & 1 & 5 \end{pmatrix}$$ - Reduced Row Echelon Form (REF): $$\begin{pmatrix} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 5 \end{pmatrix}$$ --- ### e) Given matrix: $$\begin{pmatrix} 2 & 3 & -2 & -3 & -4 & 5 \\ 4 & -2 & -3 & 2 & 1 & -2 \\ 3 & 2 & -2 & 5 & 1 & 1 \\ 2 & -5 & -3 & -3 & 1 & -1 \end{pmatrix}$$ - Row Echelon Form (EF): $$\begin{pmatrix} 2 & 3 & -2 & -3 & -4 & 5 \\ 0 & -8 & -1 & 8 & 3 & -12 \\ 0 & 0 & -2 & 6 & 2 & 10 \\ 0 & 0 & 0 & -3 & -1 & -8 \end{pmatrix}$$ - Reduced Row Echelon Form (REF): $$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & -3/2 \\ 0 & 1 & 0 & 0 & 0 & 1/4 \\ 0 & 0 & 1 & 0 & 0 & -5 \\ 0 & 0 & 0 & 1 & 0 & 2 \end{pmatrix}$$ --- ### f) Given matrix: $$\begin{pmatrix} 1 & 2 & -1 & 1 & 1 \\ 3 & 1 & 4 & -1 & 2 \\ 2 & -1 & 5 & -2 & 3 \\ 1 & -1 & 0 & -1 & -2 \end{pmatrix}$$ - Row Echelon Form (EF): $$\begin{pmatrix} 1 & 2 & -1 & 1 & 1 \\ 0 & -5 & 7 & -4 & -1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 1 & 2 \end{pmatrix}$$ - Reduced Row Echelon Form (REF): $$\begin{pmatrix} 1 & 0 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & 2 \end{pmatrix}$$ --- Let me know if you want details or have any questions! ### 5 Relative Questions: 1. What are the properties that define a matrix's row echelon form (EF)? 2. How does one perform row reduction systematically to reach reduced row echelon form (REF)? 3. What is the significance of leading 1s in reduced row echelon form? 4. How can Gaussian elimination be used to solve linear systems of equations represented by matrices? 5. What practical applications utilize matrix reduction to EF and REF? ### Tip: To achieve the reduced row echelon form efficiently, always start by making the leading coefficient of each row 1, then use it to zero out other elements in its column.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Row Reduction
Linear Algebra
Echelon Form
Reduced Echelon Form
Formulas
Elementary row operations for matrix reduction
Theorems
Gaussian Elimination
Gauss-Jordan Elimination
Suitable Grade Level
University Level (Linear Algebra Course)
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