Math Problem Statement

Which of the following augmented matrices are inconsistent? Select ALL that apply.

⎡⎣⎢⎢⎢⎢25006000008000−88−56−4−6⎤⎦⎥⎥⎥⎥

⎡⎣⎢⎢⎢⎢4−80008007300⎤⎦⎥⎥⎥⎥

⎡⎣⎢⎢⎢⎢80164−70−1402060⎤⎦⎥⎥⎥⎥

⎡⎣⎢⎢⎢⎢800000500000000−8−9445⎤⎦⎥⎥⎥⎥

Solution

To determine which of these augmented matrices are inconsistent, we need to look for rows that represent contradictions, such as a row where all coefficients of the variables are zero but the last entry (constant) is nonzero. This indicates no solution exists.

Let’s analyze each augmented matrix:

Matrix 1:

2 & 5 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 8 \\ 0 & 0 & 0 & -8 & -6 \\ -4 & -6 & 0 & 0 & 0 \\ \end{pmatrix}$$ - Row 2: $$[ 0 \, 0 \, 0 \, 0 | 8 ]$$ represents $$0 = 8$$, which is a contradiction. This row indicates that the system has no solution. - **Conclusion:** This matrix is **inconsistent**. ### Matrix 2: $$\begin{pmatrix} 4 & -8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 7 & 3 \\ 0 & 0 & 7 & 0 & 0 \\ \end{pmatrix}$$ - There is no row representing a contradiction. The system seems solvable. - **Conclusion:** This matrix is **consistent**. ### Matrix 3: $$\begin{pmatrix} 8 & 0 & 1 & 6 \\ 4 & -7 & 0 & -14 \\ 0 & 20 & 6 & 0 \\ \end{pmatrix}$$ - No row represents a contradiction. - **Conclusion:** This matrix is **consistent**. ### Matrix 4: $$\begin{pmatrix} 8 & 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & -8 & -9 & 4 & 5 \\ \end{pmatrix}$$ - Row 2: $$[ 0 \, 0 \, 0 \, 0 | 0 ]$$ represents $$0 = 0$$, which is true but doesn't affect consistency. - There are no contradictions in the other rows, so no inconsistency. - **Conclusion:** This matrix is **consistent**. ### Final Answer: The **inconsistent matrix** is: - Matrix 1. Would you like any further explanation or help with similar problems? --- ### Further Questions: 1. How do you perform Gaussian elimination on augmented matrices? 2. Can an inconsistent system have a unique solution in any case? 3. How do you identify the rank of a matrix? 4. What does it mean for a system to be consistent or inconsistent in terms of linear dependence? 5. Can a system of equations have more than one solution even if it is consistent? ### Tip: If you encounter an augmented matrix with a row that looks like $$[0 \, 0 \, 0 | b]$$ where $$b \neq 0$$, the system is automatically inconsistent because it implies a false statement like $$0 = b$$.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Augmented Matrices
Matrix Inconsistency

Formulas

Row echelon form
Contradictory row identification

Theorems

Rank of a matrix
Consistency of a system of linear equations

Suitable Grade Level

Grades 11-12

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