Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{rr} x-y+z= & -4 \\ 2 x-3 y+4 z= & -15 \\ 5 x+y-2 z= & 12 \end{array}\right. $$

Short answer

Perform row operations to get unique solutions.

Step-by-step solution

- Write the augmented matrix

Convert the given system of equations into an augmented matrix. The system is: Augmented Matrix:

- Row Reduction

Use row operations to transform the augmented matrix into its reduced row echelon form (RREF). Perform elementary row operations: - Subtract twice the first row from the second row. - Subtract Continue with these operations until the matrix is in RREF.

- Interpret the final matrix

Once the augmented matrix is in RREF, interpret it to find the solutions to the system of equations. Checks the final form to determine if a unique solution, no solution, or infinitely many solutions exist.

Key concepts

Augmented Matrix

An augmented matrix is an essential tool when solving systems of equations using matrices. It combines the coefficient matrix with the constants from the equations’ right-hand sides. To construct an augmented matrix from a system of equations:
- Write down the coefficients of each variable in a matrix form.
- Append the constants from the equations' right sides as an additional column.
This matrix visually represents the entire system of equations and makes it easier to apply row operations.

Row Operations

Row operations are crucial for simplifying matrices. They help transform the augmented matrix into a more manageable form. There are three types of elementary row operations you can perform:
- Swapping two rows.
- Multiplying a row by a non-zero constant.
- Adding or subtracting a multiple of one row to another row.
These operations can be recursively applied to gradually simplify the matrix without altering the system's solutions.

Reduced Row Echelon Form (RREF)

The goal of row operations is to convert the augmented matrix into its reduced row echelon form (RREF). An RREF matrix has these properties:
- Each leading entry in a row is 1, and it's the only nonzero entry in its column.
- The leading 1 in each row appears to the right of the leading 1 in the row above it.
- Rows with all zero entries, if any, are at the bottom.
Once in RREF, the system's solutions become evident and easier to interpret.

Elementary Row Operations

Elementary row operations are the fundamental steps used in the row reduction process. They are:
- Row Switching: Interchanging two rows, e.g., swapping row 1 and row 2.
- Row Multiplying: Multiplying all elements of a row by a nonzero scalar, e.g., multiplying row 1 by 2.
- Row Addition/Subtraction: Adding or subtracting the elements of one row to/from another, e.g., subtracting twice the first row from the second row.
These operations systematically transform the augmented matrix, guiding it towards the reduced row echelon form.