Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{lll}-1 & 1 & 4 \\ -2 & 1 & 1\end{array}\right]$$

Short answer

The final reduced row echelon form is \( \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 3 \end{bmatrix}.\)

Step-by-step solution

Write the Matrix

Write the matrix from the problem: \[\begin{bmatrix} -1 & 1 & 4 \ -2 & 1 & 1 \end{bmatrix}\] This matrix has two rows and three columns.

Create a Zero Below the First Pivot

Perform a row operation to eliminate the value below the pivot in the first column. We can add twice the first row to the second row to achieve this: - \(R_2 = R_2 + 2R_1\)New matrix:\[\begin{bmatrix} -1 & 1 & 4 \ 0 & 3 & 9 \end{bmatrix}\]

Scale the Second Row to Make the Pivot 1

Scale the second row to make the second pivot equal to 1 by dividing the entire row by 3:- \(R_2 = \frac{1}{3}R_2\)New matrix:\[\begin{bmatrix} -1 & 1 & 4 \ 0 & 1 & 3 \end{bmatrix}\]

Eliminate Above the Second Pivot

To create a zero above the second pivot, modify the first row by subtracting the second row:- \(R_1 = R_1 - R_2\)New matrix:\[\begin{bmatrix} -1 & 0 & 1 \ 0 & 1 & 3 \end{bmatrix}\]

Scale First Row to Make the Pivot 1

Finally, scale the first row to make the first pivot equal to 1 by multiplying the entire row by -1:- \(R_1 = -1 \times R_1\)Final matrix:\[\begin{bmatrix} 1 & 0 & -1 \ 0 & 1 & 3 \end{bmatrix}\] This is the reduced row-echelon form of the matrix.

Key concepts

Reduced Row Echelon Form

When we talk about matrices, the reduced row echelon form (RREF) is a special form that simplifies solving systems of linear equations. In RREF, each leading entry (also known as a pivot) is 1, and is the only non-zero entry in its column. Moreover, each pivot is to the right of the pivot in the row above it, and rows consisting entirely of zeros are at the bottom.
This form is incredibly useful because it allows us to easily read off the solutions to a system of equations. If a matrix is in RREF, the solutions can often be immediately extracted by inspection. This is especially useful in larger systems where row operations might lead to confusion. Understanding RREF is a fundamental skill for solving linear systems efficiently.

Matrix Operations

Matrix operations are the calculations you perform to transform and manipulate matrices. There are several types of operations:

• Addition and Subtraction: These are performed element-wise between matrices of the same dimensions.
• Scalar Multiplication: Each element of a matrix is multiplied by a scalar value.
• Matrix Multiplication: This involves taking the dot product of the rows of the first matrix with the columns of the second matrix.

In the context of Gaussian elimination, these operations are crucial for transforming a matrix into its reduced row echelon form. When you develop a good understanding of matrix operations, you're able to manipulate and solve complex linear algebra problems with ease. Each operation follows a specific set of rules which keeps these manipulations structured and predictable.

Row Operations

Row operations are the tools we use to transform matrices, particularly when performing Gaussian elimination. There are three primary types of row operations:

• Row Switching: Swap the positions of two rows.
• Row Multiplying: Multiply all elements of a row by a non-zero scalar.
• Row Addition: Add a multiple of one row to another row to produce a new row.

These operations are fundamental because they allow for the alteration of a matrix's form without changing the solutions of the corresponding system of equations. Each operation is reversible, ensuring that the integrity of the system is maintained. In practice, these operations are performed in a specific sequence to systematically solve for the row-echelon and reduced row-echelon forms of a matrix.