Math Problem Statement
Find rank of matrix 1 2 3 4,-2-3 1 2,-3-4 5 8,1 3 10 14
Solution
We are tasked with finding the rank of the given matrix:
1 & 2 & 3 & 4 \\ -2 & -3 & 1 & 2 \\ -3 & -4 & 5 & 8 \\ 1 & 3 & 10 & 14 \end{pmatrix}$$ ### Step 1: Form the augmented matrix The matrix is: $$A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ -2 & -3 & 1 & 2 \\ -3 & -4 & 5 & 8 \\ 1 & 3 & 10 & 14 \end{pmatrix}$$ ### Step 2: Perform Gaussian elimination We aim to transform the matrix into row echelon form (REF) by performing row operations. - **Row 2 (R2)**: Add $$2 \times R1$$ to R2: $$R2 = (-2 + 2, -3 + 4, 1 + 6, 2 + 8) = (0, 1, 7, 10)$$ - **Row 3 (R3)**: Add $$3 \times R1$$ to R3: $$R3 = (-3 + 3, -4 + 6, 5 + 9, 8 + 12) = (0, 2, 14, 20)$$ - **Row 4 (R4)**: Subtract $$1 \times R1$$ from R4: $$R4 = (1 - 1, 3 - 2, 10 - 3, 14 - 4) = (0, 1, 7, 10)$$ The resulting matrix is: $$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 7 & 10 \\ 0 & 2 & 14 & 20 \\ 0 & 1 & 7 & 10 \end{pmatrix}$$ ### Step 3: Eliminate entries below the pivot in column 2 - **Row 3 (R3)**: Subtract $$2 \times R2$$ from R3: $$R3 = (0 - 0, 2 - 2, 14 - 14, 20 - 20) = (0, 0, 0, 0)$$ - **Row 4 (R4)**: Subtract $$1 \times R2$$ from R4: $$R4 = (0 - 0, 1 - 1, 7 - 7, 10 - 10) = (0, 0, 0, 0)$$ The matrix is now: $$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 7 & 10 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ ### Step 4: Determine the rank The matrix is in row echelon form, and there are 2 non-zero rows. Therefore, the **rank of the matrix is 2**. --- Would you like more details or have any questions? Here are 5 related questions to expand on this topic: 1. How do you compute the rank of a matrix using determinants? 2. What is the significance of the rank in solving systems of linear equations? 3. How does Gaussian elimination relate to finding the null space of a matrix? 4. Can the rank of a matrix change under certain operations, like transposing? 5. How can the rank be used to check if a set of vectors is linearly independent? **Tip**: Always aim to reduce matrices to row echelon form to simplify rank determination!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Gaussian Elimination
Rank of a Matrix
Formulas
Gaussian elimination to row echelon form
Rank is the number of non-zero rows in row echelon form
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate or Advanced High School (Grades 11-12)