Problem 31 - Precalculus

Question

Perform the indicated row operations on each augmented matrix. $$\left[\begin{array}{rr|r} 1 & -2 & -3 \\ 2 & 3 & -1 \end{array}\right] \quad R_{2}-2 R_{1} \rightarrow R_{2}$$

Step-by-Step Solution

Answer

After the operation, the matrix becomes: \[ \begin{bmatrix} 1 & -2 & | & -3 \\ 0 & 7 & | & 5 \end{bmatrix} \]

Step 1: Understanding Row Operations

In this exercise, we need to perform a row operation on the given augmented matrix. The operation specified is \( R_{2} - 2R_{1} \rightarrow R_{2} \). This means we will replace the second row \( R_{2} \) with the result of subtracting twice the first row \( 2R_{1} \) from it.

Step 2: Calculate the First Element of New R2

Identify the elements in \( R_1 \) and \( R_2 \) from the original matrix: \( R_1 = [1, -2 | -3] \) and \( R_2 = [2, 3 | -1] \). Calculate the first element of the new \( R_2 \) by subtracting two times the first element of \( R_1 \) from the first element of \( R_2 \): \[ 2 - 2 \times 1 = 2 - 2 = 0 \]

Step 3: Calculate the Second Element of New R2

Proceed by calculating the second element of the new \( R_2 \) by subtracting twice the second element of \( R_1 \) from the second element of \( R_2 \): \[ 3 - 2 \times (-2) = 3 + 4 = 7 \]

Step 4: Calculate the Third Element of New R2

Finally, calculate the third element (constant from the augmented part) of the new \( R_2 \) by subtracting twice the third element of \( R_1 \) from the third element of \( R_2 \):\[ -1 - 2 \times (-3) = -1 + 6 = 5 \]

Step 5: Form the New Matrix

After performing the calculations, the new second row \( R_2 \) becomes \([0, 7 | 5]\). Therefore, the modified matrix is: \[ \begin{bmatrix}1 & -2 & | & -3 \0 & 7 & | & 5\end{bmatrix}\]

Key Concepts

Augmented Matrix

An augmented matrix is a way to represent a system of linear equations. It's essentially a matrix that includes the coefficients of the variables along with the constants from the equations. For example, consider the following system of linear equations:

• Equation 1: \( x - 2y = -3 \)
• Equation 2: \( 2x + 3y = -1 \)

To express these in an augmented matrix form, combine the coefficients and constants to form:\[\left[ \begin{array}{rr|r} 1 & -2 & -3 \ 2 & 3 & -1 \end{array} \right]\] Here, the vertical bar separates the coefficient matrix from the column of constants. Augmented matrices are a compact and efficient way to manipulate systems of equations, especially when using matrix operations to solve them.
Understanding augmented matrices is crucial for performing further operations like those in the Gaussian Elimination method.

Matrix Row Reduction

Matrix row reduction, often referred to as row echelon form or reduced row echelon form, is a technique used to simplify matrices for solving linear systems. The goal is to make the matrix easier to interpret or use for further calculation. This process is fundamental in linear algebra.

• Start with elementary row operations to create zeros below pivots (non-zero leading coefficients).
• Your final aim is to form a triangular matrix using these operations, making it simpler to back-substitute values.

For instance, the exercise provided involves transforming the second row in the given matrix, as one of the steps towards simplifying it:
Transform \( R_2 \) by subtracting \( 2R_1 \) from it.
Perform operations to zero out specific elements, while following the rules of linear algebra. This is key in solving the equations represented by the matrix
Matrix row reduction streamlines data and helps in analytical deduction by removing extraneous information.

Elementary Row Operations

Elementary row operations include simple changes we apply to rows within a matrix to either solve system equations or bring the matrix to a desired form, like the reduced row echelon form.
There are three main types of row operations:

• **Row swapping:** Exchanging two rows.
• **Row scaling:** Multiplying all elements of a row by a non-zero constant.
• **Row addition:** Adding or subtracting the multiple of one row from another row.

In the exercise provided, row addition is used: \( R_{2} - 2R_{1} \rightarrow R_{2} \). This modifies the second row by subtracting twice the first row from it. This operation is fundamental in shaping the matrix towards a simpler form, such as making zeros beneath pivot positions.
Such transformations do not change the solution of the corresponding system of equations, preserving its integrity while easing further calculations.