Augmented Matrices

Augmented Matrices

An augmented matrix is a matrix that represents a system of linear equations. It is a convenient way to solve systems of equations using matrix methods, such as Gaussian elimination or row reduction. The augmented matrix includes both the coefficients of the variables and the constants from the equations, combined into a single matrix.

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1. Definition of Augmented Matrix

Given a system of linear equations:

$$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b_1$$ $$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b_2$$ $$\vdots$$ $$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b_m$$

where $x_1, x_2, \dots, x_n$ are the variables, $a_1, a_2, \dots, a_n$ are the coefficients of the variables, and $b_1, b_2, \dots, b_m$ are the constants, the augmented matrix for this system is written as:

$$\left[ \begin{array}{ccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array} \right]$$

The augmented matrix combines the coefficient matrix and the column of constants into a single matrix. The vertical bar $|$ separates the coefficients of the variables from the constants.

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2. Structure of Augmented Matrices

The augmented matrix has the following general form:

$$\left[ \begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array} \right]$$

• The left side of the matrix contains the coefficients of the variables (the original coefficient matrix).
• The right side (after the bar) contains the constants from the right-hand side of the equations.

Example:

For the system of equations:

$$2x + 3y = 5$$ $$4x - y = 3$$

The augmented matrix would be:

$$\left[ \begin{array}{cc|c} 2 & 3 & 5 \\ 4 & -1 & 3 \\ \end{array} \right]$$

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3. Solving Systems Using Augmented Matrices

The augmented matrix is primarily used in methods like Gaussian elimination and Gauss-Jordan elimination to solve systems of linear equations.

Gaussian Elimination:

This method involves performing row operations on the augmented matrix to reduce it to row echelon form (REF). The goal is to simplify the system so that you can solve for the variables.

1. Row Swaps: Swap two rows.
2. Scalar Multiplication: Multiply a row by a nonzero scalar.
3. Row Addition/Subtraction: Add or subtract a multiple of one row to/from another.

After performing these row operations, the matrix is simplified into an upper triangular form, from which you can back-substitute to find the solutions.

Gauss-Jordan Elimination:

This is an extension of Gaussian elimination where the matrix is further reduced to reduced row echelon form (RREF). The goal is to get a matrix where the leading entry (pivot) in each row is 1, and all other entries in the column of a pivot are zero. This allows you to immediately read off the solutions for the variables.

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4. Example of Solving Using Augmented Matrices

Consider the following system of equations:

$$2x + 3y = 5$$ $$4x - y = 3$$

The corresponding augmented matrix is:

$$\left[ \begin{array}{cc|c} 2 & 3 & 5 \\ 4 & -1 & 3 \\ \end{array} \right]$$

We will use Gaussian elimination to solve this system.

1. Step 1: Make the leading coefficient of the first row equal to 1:

   Divide the first row by 2:

   $$\left[ \begin{array}{cc|c} 1 & 1.5 & 2.5 \\ 4 & -1 & 3 \\ \end{array} \right]$$

2. Step 2: Eliminate the $x$-term from the second row:

   Subtract 4 times the first row from the second row:

   $$\left[ \begin{array}{cc|c} 1 & 1.5 & 2.5 \\ 0 & -7 & -7 \\ \end{array} \right]$$

3. Step 3: Make the leading coefficient in the second row equal to 1:

   Divide the second row by $-7$:

   $$\left[ \begin{array}{cc|c} 1 & 1.5 & 2.5 \\ 0 & 1 & 1 \\ \end{array} \right]$$

4. Step 4: Eliminate the $y$-term from the first row:

   Subtract 1.5 times the second row from the first row:

   $$\left[ \begin{array}{cc|c} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{array} \right]$$

Now, the system is in reduced row echelon form (RREF), and we can read off the solution:

$$x = 1, \quad y = 1$$

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5. Special Notes

• Augmented matrix operations are equivalent to performing operations on the original system of equations.
• The row operations on the augmented matrix correspond to equivalent operations on the system of equations, meaning that the solution to the matrix will be the same as the solution to the system of equations.

Conclusion

An augmented matrix is a compact and efficient representation of a system of linear equations. It allows you to use matrix operations like Gaussian elimination and Gauss-Jordan elimination to find solutions. By manipulating the augmented matrix, you can solve systems of equations systematically and efficiently.