Problem 16 - College Algebra

Question

perform each matrix row operation and write the new matrix. $$ \left[\begin{array}{rrr|r} 1 & -1 & 5 & -6 \\ 3 & 3 & -1 & 10 \\ 1 & 3 & 2 & 5 \end{array}\right]-3 R_{1}+R_{2} $$

Step-by-Step Solution

Answer

The new matrix, after performing the row operation -3R1+R2, is \(\left[\begin{array}{ccc|c} 1 & -1 & 5 & -6 \ -3 & 6 & -16 & 28 \ 1 & 3 & 2 & 5 \end{array}\right]\)

Step 1: Identify Matrix and Operation

The matrix to work with is \(\left[\begin{array}{rrr|r}1 & -1 & 5 & -6 \ 3 & 3 & -1 & 10 \ 1 & 3 & 2 & 5\end{array}\right]\) and the row operation to perform on this matrix is -3 times the first row add to the second row, represented as -3R1+R2.

Step 2: Perform the Row Operation

The row operation -3R1+R2 means to multiply every element in the first row by -3, and then adding it to the corresponding element in the second row. Doing this, the second row will become: -3(1)+3=-3, -3*(-1)+3=6, -3*5-1=-16, -3*(-6)+10=28.

Step 3: Write the New Matrix

The new matrix, after performing the row operation, will be:\(\left[\begin{array}{ccc|c} 1 & -1 & 5 & -6 \ -3 & 6 & -16 & 28 \ 1 & 3 & 2 & 5 \end{array}\right]\)

Key Concepts

Elementary Row Operations

Elementary row operations are useful tools in matrix algebra. These operations can change a matrix into a more usable form without altering the solutions of the system of equations it represents.
They are essential for solving systems of linear equations and include three main types:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a scalar multiple of one row to another row.

These operations help simplify matrices, making them easier to manipulate and solve. For example, by using elementary row operations, one can transform a matrix into its row-echelon form or even its reduced row-echelon form. This helps in finding solutions to linear systems more effectively. By performing such operations correctly, the integrity of the original equations is preserved, guaranteeing that any solution obtained from the modified matrix also satisfies the original system.

Augmented Matrix

An augmented matrix is a powerful concept for solving systems of linear equations. It is a matrix form that combines the coefficients of a system of equations with their constant terms in a single matrix.

• The left side of the matrix represents the coefficients of the variables.
• The right side, separated by a vertical line, represents the constants from the equations.

For the given problem, the initial augmented matrix is:\[\left[\begin{array}{rrr|r}1 & -1 & 5 & -6 \3 & 3 & -1 & 10 \1 & 3 & 2 & 5\end{array}\right]\]This form allows for straightforward application of elementary row operations to manipulate and solve the system. Once transformed adequately, the matrix can reveal the solutions to the system through back-substitution or direct reading from its reduced form.

Linear Equations

Linear equations are mathematical expressions representing lines in a standard coordinate system. They take the form of combinations of variables each multiplied by a constant and summed to equal a constant. These equations are fundamental to algebra and are often used to model real-life situations.
Linear systems can be represented conveniently in matrix form, allowing for systematic solutions through various methods, such as:

• Gaussian elimination
• Matrix inversion (for square matrices)
• Cramer's rule (for square systems with a non-zero determinant)

By understanding linear equations as rows in a matrix, one can apply techniques like elementary row operations to find solutions efficiently. The key is to manipulate the matrix without altering the inherent relationships defined by the equations. This is why operational transformations like the ones in the problem are crucial—they maintain the integrity of the original equations while simplifying their resolution.

Matrix Transformations

Matrix transformations involve changing a matrix from one form to another to reveal solutions or further insights. These transformations can simplify systems represented by the matrix or make computations easier.
The transformation of matrices using row operations can lead to forms such as:

• Row-echelon form
• Reduced row-echelon form (also known as row-reduced form)

In our exercise, the row operation \[-3R_1 + R_2\] transforms the initial matrix by altering its second row. This transformation does not change the solution set of the system but makes it easier to solve by reducing complexity. The resultant matrix is:\[\left[\begin{array}{ccc|c}1 & -1 & 5 & -6 \-3 & 6 & -16 & 28 \1 & 3 & 2 & 5\end{array}\right]\]This kind of manipulation is essential for computational efficiency and understanding the structural essence of systems of linear equations.