Math Problem Statement

solve

Solution

To solve the problem presented in the image, I will first transcribe the content and then proceed with solving the problem. The image seems to involve a mathematical matrix calculation.

Here is the transcription of the problem:

Problem: "Use elementary row operations to reduce the matrix to echelon form and hence determine the rank of the matrix. The system is consistent or inconsistent, which of the statements below describes the system?"

Matrix: [121242121]\begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{bmatrix}

Let's proceed step-by-step:

  1. Identify the matrix to be reduced:
1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{bmatrix}$$ 2. **Perform elementary row operations to reduce it to echelon form.** First, let's start with the given matrix: $$\begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{bmatrix}$$ We start by using the first row to eliminate the entries below the first pivot. - $$R2$$ = $$R2$$ - 2$$R1$$ - $$R3$$ = $$R3$$ - $$R1$$ $$\begin{bmatrix} 1 & 2 & 1 \\ 2-2(1) & 4-2(2) & 2-2(1) \\ 1-1(1) & 2-1(2) & 1-1(1) \end{bmatrix} = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ Since the second and third rows are all zeros, we already have the echelon form of the matrix. 3. **Determine the rank of the matrix:** The rank of a matrix is the number of leading 1's in the echelon form of the matrix. Here, there is only one leading 1 in the echelon form matrix. Hence, the rank of the matrix is 1. 4. **Consistency of the system:** To determine if the system is consistent or inconsistent, we look at the augmented matrix. If the matrix were augmented with a column of constants, say: $$\begin{bmatrix} 1 & 2 & 1 & | b_1 \\ 2 & 4 & 2 & | b_2 \\ 1 & 2 & 1 & | b_3 \end{bmatrix}$$ We see that the second and third rows are all zeros, implying there would be no pivot in these rows for any non-zero entries in the augmented column. This means the system is consistent if $$b_2 = 0$$ and $$b_3 = 0$$. If these conditions hold, the system is consistent. Would you like any further details or have any questions? Here are some related questions: 1. What is the definition of matrix rank? 2. How do elementary row operations affect the determinant of a matrix? 3. What is the significance of the leading 1s in row echelon form? 4. How can you determine if a matrix is in row echelon form? 5. What are the possible solutions to a consistent linear system? 6. How do you determine if a system of equations is homogeneous? 7. What are the properties of a matrix with rank 1? 8. How do you perform row reduction on a matrix with more rows or columns? **Tip:** Understanding the concept of linear independence is crucial when dealing with matrix ranks and solutions to systems of linear equations.

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
Row Echelon Form
System of Linear Equations

Formulas

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Theorems

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Suitable Grade Level

University Level