Problem 18 - Algebra and Trigonometry

Question

Perform each matrix row operation and write the new matrix. $$ \left[\begin{array}{rrrr|r} {1} & {-5} & {2} & {-2} & {4} \\ {0} & {1} & {-3} & {-1} & {0} \\ {3} & {0} & {2} & {-1} & {6} \\ {-4} & {1} & {4} & {2} & {-3} \end{array}\right] \begin{array}{r} {-3 R_{1}+R_{3}} \\ {4 R_{1}+R_{4}} \end{array} $$

Step-by-Step Solution

Answer

The resulting matrix after performing the operations is \[\begin{array}{cccc|c}1 & -5 & 2 & -2 & 4 \0 & 1 & -3 & -1 & 0 \0 & 15 & -4 & 5 & -6 \0 & -19 & 12 & 6 & 13\end{array}\]

Step 1 - Write Down the Matrix

The first task is writing down the matrix. This is it:\[\begin{array}{cccc|c}1 & -5 & 2 & -2 & 4 \0 & 1 & -3 & -1 & 0 \3 & 0 & 2 & -1 & 6 \-4 & 1 & 4 & 2 & -3\end{array}\]

Step 2 - Perform the Operations on the Rows

Now, perform the operations on the relevant rows. The required operations are -3 times the first row added to the third row (-3 R1 + R3) and 4 times the first row added to the fourth row (4R1 + R4). This results in the new matrix, shown below:\[\begin{array}{cccc|c}1 & -5 & 2 & -2 & 4 \0 & 1 & -3 & -1 & 0 \0 & 15 & -4 & 5 & -6 \0 & -19 & 12 & 6 & 13\end{array}\]

Key Concepts

Elementary Row Operations

Elementary row operations are fundamental techniques used in linear algebra to simplify matrices and solve systems of linear equations. These operations can change a matrix into a simpler form without altering the solutions of the equation it represents. The three types of elementary row operations are:

• Row swapping: Exchanging two rows within a matrix.
• Row multiplication: Multiplying all elements in a row by a non-zero scalar.
• Row addition: Adding or subtracting the multiple of one row to another row.

These operations facilitate transforming matrices into forms that are easier to solve, such as row-echelon form or reduced row-echelon form. It is crucial to remember that although the appearance of the matrix may change, the set of solutions remains consistent throughout the process.

Augmented Matrix

An augmented matrix is a compact and convenient way of representing a system of linear equations. It combines the coefficients of the variables and the constants from the system's equations into a single matrix. This allows you to handle and solve systems of equations efficiently using techniques such as row operations and Gaussian elimination.
The construction of an augmented matrix involves the following steps:

• Write the coefficients of each variable in a row for each equation.
• Include an additional column to the right for the constants from each equation, separated by a vertical line.

This structure provides a streamlined approach to manipulating the equations collectively. The augmented matrix thus simplifies managing the complex operations necessary for finding solutions.

System of Linear Equations

A system of linear equations consists of multiple equations, all of which must be solved together because they share variables. Such systems can model various real-world situations, like network flows, blending problems, or optimization tasks. Each equation in the system can be written in matrix form, leading to an augmented matrix representation.
The goal is typically to find values for the variables that satisfy all equations simultaneously. Two key methods for solving these systems include:

• Substitution and Elimination: Solving one equation for a variable and substituting it into another.
• Matrix Methods: Using augmented matrices, Gaussian elimination, or matrix inversion.

Understanding and solving systems of linear equations are essential skills in mathematics, engineering, and various scientific fields.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations through elementary row operations. By applying these operations, one can transform any augmented matrix into an upper triangular form or even a reduced row-echelon form, making it easier to solve.
The procedure involves:

• Applying row operations to systematically eliminate coefficients below the leading 1 in each row.
• Continuing this process across all rows until a triangular form is achieved.

Once in this form, the system can easily be solved by back-substitution. Gaussian elimination is not just a powerful tool for solving systems of equations but also serves as a foundation for deeper insights into the properties of matrices, such as determining ranks or calculating determinants.