https://file.aiquickdraw.com/mathbot/file/1726721590008al6nbqeu.jpg

Let's analyze the augmented matrix given in the image and solve the questions accordingly.

The augmented matrix is:

\[
\begin{bmatrix}
1 & 1 & 0 & 2 \\
1 & 2 & 0 & 1 \\
3 & 5 & a & b
\end{bmatrix}
\]

We need to solve the following:

### (a) For what values of \(a\) will the system have a unique solution? What is the solution?
### (b) For what values of \(a\) and \(b\) will the system have infinitely many solutions?
### (c) For what values of \(a\) and \(b\) will the system be inconsistent?

### Steps to proceed:

We will perform row reduction to analyze the system of equations represented by the augmented matrix. The system has three variables and three equations. To understand the behavior of the system, let's row reduce and consider the conditions that make the system have unique, infinite, or no solutions.

#### Step 1: Row Reduction
Let me begin by reducing the matrix step by step.It seems that we encountered a type issue due to the symbolic variables \( a \) and \( b \). I'll need to treat \( a \) and \( b \) symbolically to properly carry out the row operations. Let me handle this by applying symbolic computation.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you would like, I can guide you step by step through manually reducing the augmented matrix to solve the system for unique, infinite, or inconsistent solutions. Let me know if you'd prefer that!

Consider a linear system whose augmented matrix is of the form [1 1 0 2; 1 2 0 1; 3 5 a b]. For what values of a and b will the system have: (a) a unique solution? (b) infinitely many solutions? (c) be inconsistent?

This detailed solution involves analyzing a linear system with an augmented matrix to find conditions for unique solutions, infinitely many solutions, or inconsistency. The problem includes Gaussian elimination, row reduction, and understanding linear algebra concepts such as the Rouché–Capelli theorem.

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