https://file.aiquickdraw.com/mathbot/file/17260993443411qbq7t1b.jpg

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:

\[
\begin{pmatrix}
1 & 1 & 3 & | & 2 \\
1 & 2 & 4 & | & 3 \\
1 & 3 & a & | & b
\end{pmatrix}
\]

### (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.

Consider a linear system whose augmented matrix is of the form: [ [1, 1, 3 | 2], [1, 2, 4 | 3], [1, 3, a | b] ]. (a) For what values of a and b will the system have infinitely many solutions? (b) For what values of a and b will the system be inconsistent?

Learn how to find the values of a and b in a linear system represented by an augmented matrix. Discover the conditions for infinitely many solutions and inconsistency using Gaussian elimination and row reduction techniques. Suitable for students in advanced high school or early university courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation