Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{cc}-5 & 7 \\ 10 & 14\end{array}\right]$$

Short answer

The matrix is transformed into the identity matrix \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\).

Step-by-step solution

Formulate Augmented Matrix

The matrix is given as \( \begin{bmatrix}-5 & 7 \ 10 & 14 \end{bmatrix} \). Since this is a 2x2 matrix, we will start treating it as an augmented matrix for simplicity, where no extra variables from an independent term column are needed.

Make Leading Coefficient of First Row A 1

To create a leading 1 in the first row, first column, we multiply the entire first row by \(-\frac{1}{5}\). This produces a new matrix: \[\begin{bmatrix} 1 & -\frac{7}{5} \ 10 & 14 \end{bmatrix}\].

Eliminate First Element of Second Row

In order to eliminate the first element of the second row, make it zero by subtracting 10 times the first row from the second row: \[ R_2 = R_2 - 10R_1 \].This results in the matrix: \[\begin{bmatrix} 1 & -\frac{7}{5} \ 0 & 28 \end{bmatrix}\].

Make the Leading Coefficient of the Second Row A 1

Divide the entire second row by 28 to make the leading coefficient 1. This modifies the matrix to: \[\begin{bmatrix} 1 & -\frac{7}{5} \ 0 & 1 \end{bmatrix}\].

Eliminate Non-Zero Entry Above Leading 1 of Second Row

To create zero above the leading 1 of the second row, update the first row using its second row. Adjust it as follows: \( R_1 = R_1 + \frac{7}{5}R_2 \). This gives:\[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\], which is the identity matrix.

Key concepts

Reduced Row Echelon Form

The Reduced Row Echelon Form (RREF) of a matrix is a specialized form that makes solving linear equations straightforward. In RREF:

• Each leading coefficient (the first nonzero number from the left, in a row) is 1.
• Each column that contains a leading 1 has zeros in all its other entries.
• The leading 1 in each row appears to the right of the leading 1 in the row above it.
• Any row containing only zeros is pushed down to the bottom.

In simpler terms, RREF structures a matrix in a way that clearly shows the solutions to linear equations. When using Gaussian Elimination, each step aims to transform the matrix into RREF through elementary row operations like row swapping, scaling rows, and adding or subtracting rows. This method is powerful in reducing complex systems of equations into more manageable forms.
Understanding RREF can make it easier for you to identify solutions directly from the matrix, improving how efficiently you can solve algebraic problems without computations getting lost in the process.

Matrix Algebra

Matrix Algebra is the study of matrices and their operations. This algebra is crucial in many fields such as physics, computer science, and economics. With matrices, complex calculations such as linear transformations and systems of linear equations can be simplified. Sub-operations include:

• Addition and Subtraction: Done element by element for matrices of the same size.
• Multiplication: Involves combining rows of the first matrix with columns of the second matrix.
• Scalar Multiplication: Multiplying every element by a scalar (numerical constant).
• Transpose: Flipping a matrix over its diagonal, turning its rows into columns and vice versa.

Gaussian Elimination, a popular method in Matrix Algebra, systematically applies row operations to simplify a matrix or solve systems of equations. By becoming familiar with Matrix Algebra, you enhance your ability to work through various mathematical models and challenges confidently.

Identity Matrix

An Identity Matrix is a special kind of square matrix that plays the same role in matrix multiplication as the number 1 plays in regular arithmetic. In an identity matrix:

• All elements along the main diagonal are 1s.
• All other elements in the matrix are 0s.

When you multiply any matrix by an identity matrix, the original matrix remains unchanged. This property makes the identity matrix a crucial tool in matrix algebra and linear transformations. During Gaussian Elimination, the goal is often to transform a given matrix into an identity matrix through row operations. Achieving this indicates that the system of equations represented by the matrix has been fully solved, providing the solutions directly. Understanding the role and properties of identity matrices is essential for efficiently navigating problems in matrix algebra.