91Ó°ÊÓ

An augmented matrix is a brilliant tool used in linear algebra to represent a system of linear equations. It combines the coefficient matrix of the variables and the constant values into one neat package. This makes it much easier to visualize and solve the system.

In our task, we are given:

• Matrix A: \[A = \begin{bmatrix} 1 & 2 & 1 \ -3 & -1 & 2 \ 0 & 5 & 3 \end{bmatrix}\]
• Matrix b:\[\mathbf{b} = \begin{bmatrix} 0 \ 1 \ -1 \end{bmatrix}\]

We merge matrix A with matrix b to form the augmented matrix. It's like attaching the answers to the equations to see the whole picture:\[\left[ \begin{array}{ccc|c} 1 & 2 & 1 & 0 \ -3 & -1 & 2 & 1 \ 0 & 5 & 3 & -1 \end{array} \right]\] This matrix beautifully lays out the linear system for easy manipulation.

The row-echelon form allows us to simplify the augmented matrix, making it much easier to extract solutions for our system of equations. This form has zeros below its pivots (the leading non-zero numbers in each row). It looks cleaner and more organized.

To achieve this, we perform operations on the rows of our augmented matrix, such as swapping rows, multiplying entire rows by non-zero constants, and adding rows together to form zeros strategically.

In our exercise, after a series of smart row operations, we obtain:\[\left[ \begin{array}{ccc|c} 1 & 2 & 1 & 0 \ 0 & 5 & 5 & 1 \ 0 & 0 & -2 & -2 \end{array} \right]\] This form is structured to allow further simplification into reduced row-echelon form, which directly reveals the solution.

A matrix equation, like \( A \mathbf{x} = \mathbf{b} \), is a compact way to represent a system of linear equations. Here, \( A \) is the matrix of coefficients, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constant vector.

This equation showcases the powerful relationship between matrices and vectors, allowing us to handle multiple equations simultaneously with elegant mathematical operations.

By transforming the matrix equation into its augmented matrix form, we employ row operations to manipulate its structure. Through simplification to reduced row-echelon form, we decipher the solutions encapsulated within.

In our example, converting \( A \mathbf{x} = \mathbf{b} \) into an augmented matrix laid the foundation for deriving the solution.

A solution vector neatly encapsulates the solutions to a system of linear equations. After reducing the augmented matrix to its simplest form, the values in the last column correspond to the solutions for each variable.

In reduced row-echelon form, the solutions become evident. Here are the solutions from our matrix:

• \( x_1 = -\frac{7}{5} \)
• \( x_2 = 0 \)
• \( x_3 = 1 \)

Thus, the solution vector is:\[\mathbf{x} = \begin{bmatrix} -\frac{7}{5} \ 0 \ 1 \end{bmatrix}\] The solution vector succinctly presents the answers in one compact form, making them simple and clear to interpret.