Math Problem Statement
Solution
The problem provided involves a system of linear equations (SPL) in matrix form and asks to determine conditions for the parameter such that:
- The system has a unique solution.
- The system has infinitely many solutions.
- The system has no solution.
The augmented matrix is given as:
\begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 1 & 3 \\ -2 & -1 & -(a+1) & a-3 \end{pmatrix}$$ ### Steps to Solve: 1. **Unique Solution**: This occurs when the matrix $$A$$ is invertible, which happens if the determinant $$\text{det}(A) \neq 0$$. 2. **Infinitely Many Solutions**: This occurs when the rank of the matrix $$A$$ is less than 3 (number of unknowns), but the rank of the augmented matrix is the same as the rank of $$A$$. 3. **No Solution**: This happens when the rank of the matrix $$A$$ is less than the rank of the augmented matrix, leading to an inconsistency. Would you like a detailed breakdown of finding the determinant and analyzing the cases for the parameter $$a$$? Let me know if you'd like to proceed! In the meantime, here are five related questions to deepen your understanding: 1. What is the determinant of matrix $$A$$ as a function of $$a$$? 2. How do we determine if a matrix is invertible based on its determinant? 3. How can we use row reduction to find the rank of a matrix? 4. What are the conditions for a system of linear equations to have infinitely many solutions? 5. How can an augmented matrix help us detect inconsistent systems? **Tip**: When determining the number of solutions to a system of linear equations, always check the rank of both the coefficient matrix and the augmented matrix!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
System of Linear Equations
Matrix Theory
Determinants
Ranks of Matrices
Formulas
Determinant of a 3x3 matrix
Rank of a matrix
Consistency of a system of linear equations
Theorems
Unique solution occurs if det(A) ≠ 0
Infinitely many solutions occur when rank(A) = rank([A|b]) < n
No solution occurs when rank(A) < rank([A|b])
Suitable Grade Level
Grades 11-12, Undergraduate
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