Question

Perform the given row operations on \(A,\) where $$A=\left[\begin{array}{ccc}2 & -1 & 7 \\ 0 & 4 & -2 \\ 5 & 0 & 3\end{array}\right]$$ $$-\frac{5}{2} R_{1}+R_{3} \rightarrow R_{3}$$

Short answer

The new matrix is \(\begin{bmatrix} 2 & -1 & 7 \\ 0 & 4 & -2 \\ 0 & \frac{5}{2} & -\frac{29}{2} \end{bmatrix}\).

Step-by-step solution

Identify the Row Operation Target

We need to perform the row operation \(-\frac{5}{2} R_{1}+R_{3} \rightarrow R_{3}\). This indicates we must multiply the first row \(R_1\) by \(-\frac{5}{2}\), then add the result to the third row \(R_3\) and replace \(R_3\) with the new values.

Multiply the First Row by the Fraction

Take the first row \(R_1 = [2, -1, 7]\) and multiply each element by \(-\frac{5}{2}\):- Element 1: \(2 \times -\frac{5}{2} = -5\)- Element 2: \(-1 \times -\frac{5}{2} = \frac{5}{2}\)- Element 3: \(7 \times -\frac{5}{2} = -\frac{35}{2}\)The result is \([-5, \frac{5}{2}, -\frac{35}{2}]\).

Add the Result to the Third Row

Now add the results from Step 2 to the third row \(R_3 = [5, 0, 3]\) of the matrix \(A\):- Element 1: \(5 + (-5) = 0\)- Element 2: \(0 + \frac{5}{2} = \frac{5}{2}\)- Element 3: \(3 + (-\frac{35}{2}) = -\frac{29}{2}\)These will be the new values for \(R_3\).

Write Down the New Matrix

Replace the third row \(R_3\) in matrix \(A\) with the new values obtained in Step 3. The resulting matrix is:\[A = \begin{bmatrix} 2 & -1 & 7 \ 0 & 4 & -2 \ 0 & \frac{5}{2} & -\frac{29}{2} \end{bmatrix}\]

Key concepts

Elementary Row Operations

Elementary row operations are essential tools in matrix algebra, helping simplify matrices and solve systems of linear equations. These operations include:

• Swapping two rows.
• Multiplying a row by a non-zero constant.
• Adding multiples of one row to another.

In the given exercise, we performed the operation that involves adding a multiple of the first row to the third row. This operation was critical in modifying the matrix while maintaining its solution set. It's important to remember that these row operations do not alter the solution to the matrix equation; therefore, they are widely used in techniques such as Gaussian elimination to solve linear systems. Understanding these operations helps you transform matrices into forms that are easier to interpret and solve.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the various operations that can be performed on them. A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The operations you can perform on matrices include:

• Addition and subtraction of matrices.
• Multiplication by a scalar.
• Matrix multiplication.
• Taking the transpose of a matrix.
• Finding the determinant and inverse of square matrices.

In our exercise, we focused on manipulating the matrix using row operations. Such operations help simplify matrices to solve systems of linear equations more effortlessly. Matrix algebra is fundamental in various scientific fields, including computer science, physics, and engineering, as it provides a compact and efficient way to deal with multiple linear equations.

Linear Equations

Linear equations are equations of first order, involving constants and linear functions of variables. These equations are foundational in algebra, often represented in the form:\[ ax + by + cz + \,\ldots = d \]where each term is either a constant or the product of a constant and a single variable.
Given multiple linear equations, these can form a system where all the equations intersect at a particular solution.
Matrices offer a straightforward way to express a system of linear equations and solve it. For instance, each row in a matrix often corresponds to a linear equation, and elementary row operations can help isolate variables, making solutions more transparent.
In our exercise, the task was part of simplifying a system of equations to identify these solutions. By mastering the use of matrix representations and transformations, solving complex linear systems becomes more manageable.