Math Problem Statement
Solution
We are given an augmented matrix representing a system of linear equations and asked to determine conditions for infinitely many solutions and inconsistency. The matrix provided is:
1 & 1 & 3 & | & 2 \\ 1 & 2 & 4 & | & 3 \\ 1 & 3 & a & | & b \end{pmatrix}$$ ### (a) Infinitely Many Solutions A system of linear equations has infinitely many solutions when there is a row of the form: $$0 \quad 0 \quad 0 \quad | \quad 0$$ This means the third row must reduce to zero for all the elements on the left and right side of the augmented matrix. #### Step 1: Row Reduction We will perform Gaussian elimination to row-reduce the matrix and simplify it. 1. **Subtract the first row from the second and third rows:** $$R_2 \to R_2 - R_1: \quad (1 \ 2 \ 4 \ | \ 3) - (1 \ 1 \ 3 \ | \ 2) = (0 \ 1 \ 1 \ | \ 1)$$ $$R_3 \to R_3 - R_1: \quad (1 \ 3 \ a \ | \ b) - (1 \ 1 \ 3 \ | \ 2) = (0 \ 2 \ a - 3 \ | \ b - 2)$$ 2. **Subtract 2 times the second row from the third row:** $$R_3 \to R_3 - 2R_2: \quad (0 \ 2 \ (a-3) \ | \ b-2) - 2(0 \ 1 \ 1 \ | \ 1) = (0 \ 0 \ a - 5 \ | \ b - 4)$$ Now the augmented matrix becomes: $$\begin{pmatrix} 1 & 1 & 3 & | & 2 \\ 0 & 1 & 1 & | & 1 \\ 0 & 0 & a - 5 & | & b - 4 \end{pmatrix}$$ #### Condition for Infinitely Many Solutions For infinitely many solutions, the third row must reduce to: $$0 \quad 0 \quad 0 \quad | \quad 0$$ Thus, we require both: $$a - 5 = 0 \quad \text{and} \quad b - 4 = 0$$ So, the system will have infinitely many solutions when: $$a = 5 \quad \text{and} \quad b = 4$$ ### (b) Inconsistent System A system is inconsistent if, after row reduction, we get a row of the form: $$0 \quad 0 \quad 0 \quad | \quad c \quad (\text{where } c \neq 0)$$ This corresponds to a contradiction like $$0 = c$$. From the reduced matrix: $$\begin{pmatrix} 1 & 1 & 3 & | & 2 \\ 0 & 1 & 1 & | & 1 \\ 0 & 0 & a - 5 & | & b - 4 \end{pmatrix}$$ If $$a - 5 = 0$$ but $$b - 4 \neq 0$$, we will have an inconsistent system. Therefore, the system is inconsistent when: $$a = 5 \quad \text{and} \quad b \neq 4$$ ### Final Answers: - **(a)** The system will have infinitely many solutions when $$a = 5$$ and $$b = 4$$. - **(b)** The system will be inconsistent when $$a = 5$$ and $$b \neq 4$$. --- Do you want details on any part of the explanation or have additional questions? Here are 5 related questions for further understanding: 1. What is Gaussian elimination and how does it work in solving linear systems? 2. How do we identify if a system has no solution or infinitely many solutions from its augmented matrix? 3. What is the difference between row-reduced echelon form (RREF) and Gaussian elimination? 4. Can a system be consistent if the augmented matrix has a row with all zeros on the left and a non-zero value on the right? 5. How do linear transformations relate to solving systems of linear equations? **Tip:** Always check the determinant of the coefficient matrix for a quick test of uniqueness of solutions. If it's zero, the system is either inconsistent or has infinitely many solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Row Reduction
Gaussian Elimination
Formulas
Row reduction
Condition for infinitely many solutions: 0 = 0
Condition for inconsistency: 0 = c where c ≠ 0
Theorems
Gaussian Elimination
Consistent vs. Inconsistent Systems
Infinitely Many Solutions Theorem
Suitable Grade Level
Grades 11-12 or early university level
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