Problem 13 - Algebra and Trigonometry

Question

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{llll}{1} & {2} & {8} & {0} \\ {0} & {1} & {3} & {2} \\\ {0} & {0} & {0} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

(a) The matrix is in row-echelon form. (b) The matrix is not in reduced row-echelon form. (c) The system is \(x_1 + 2x_2 + 8x_3 = 0\), \(x_2 + 3x_3 = 2\).

Step 1: Identify Leading Entries

Examine the matrix and identify the leading entry (first non-zero entry from the left) in each row. The given matrix is \[\begin{bmatrix}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]. The leading entry of the first row is 1, located in the first column, and the leading entry of the second row is 1, located in the second column.

Step 2: Determine Row-Echelon Form

To check if this matrix is in row-echelon form, confirm that all non-zero rows are above any rows of all zeros, the leading entry of a non-zero row is to the right of the leading entry of the row above it, and the entries below a leading entry are all zeros. The matrix satisfies all these conditions, confirming it is in row-echelon form.

Step 3: Check for Reduced Row-Echelon Form

Check if the matrix is in reduced row-echelon form by ensuring that, in addition to row-echelon form conditions, leading entries are 1, the leading entry is the only non-zero entry in its column, and every leading 1 has zeros above it. The second leading entry does not have all zeros above and below it, so the matrix is not in reduced row-echelon form.

Step 4: Write the System of Equations

Translate the augmented matrix back into the system of equations. The matrix \(\begin{bmatrix}1 & 2 & 8 & | & 0 \0 & 1 & 3 & | & 2 \0 & 0 & 0 & | & 0\end{bmatrix}\) represents the following system:1. \(x_1 + 2x_2 + 8x_3 = 0\)2. \(x_2 + 3x_3 = 2\)

Key Concepts

Understanding Matrices

A matrix is like a table filled with numbers arranged in rows and columns. Imagine it as a rectangular grid where each spot is called an element. Matrices are a super helpful tool in mathematics, especially when dealing with systems of equations.

• Rows and Columns: Just like a spreadsheet, a matrix has rows that go across and columns that go down. It’s important to get familiar with reading them since they help organize data or equations.
• Elements: Each number within a matrix is called an element. For example, in the matrix given in the exercise, 1 is an element located at position (1,1).
• Types of Matrices: Matrices can vary in size, being small like our example, or extremely large, depending on how much information they need to handle.

Matrices are essential because they allow us to perform complex calculations like solving systems of linear equations, which would be quite tedious to do by hand without the structured format matrices provide.

Solving Systems of Equations with Matrices

A system of equations is a collection of equations that need to be solved together. Each equation provides information about some quantities, and all equations must be satisfied simultaneously.

• Real-Life Application: Imagine you have several relationships in a business where you know the total sales and expenses, but need to determine individual product profits. This can be solved using systems of equations.
• Matrix Representation: Instead of writing out all equations, you can pack the information into a matrix. This makes it easier to handle and apply mathematical techniques to solve the system.
• Augmented Matrix: This type of matrix includes both the coefficients of your variables and the constants from each equation, as seen in the exercise, which helps in directly transforming the problem into matrix form.

By using matrices to represent systems of equations, the problem becomes more visual and often simpler to understand, as you can clearly see the relationships and constraints all together.

Reduced Row-Echelon Form (RREF) Made Simple

The reduced row-echelon form (RREF) of a matrix is a special form that makes it super easy to solve systems of equations. Think of it as a neat, organized way to quickly read the solution of an equation system from a matrix.

• How to Identify RREF: In RREF, each leading entry (the first non-zero number in a row) must be 1. This 1 is the only number in that column; everything else in the column should be zero.
• Step-by-Step Transformation: You can transform a regular matrix to its reduced row-echelon form using a series of row operations. These include swapping rows, multiplying a row by a non-zero number, and adding or subtracting the rows.
• Visual Clarity: In RREF, the matrix looks a bit like steps, allowing you to easily pick out solutions to the system of equations. It's much neater and helps avoid errors when solving.

By putting a matrix in reduced row-echelon form, finding the solution to a system of equations becomes straightforward and ensures that you aren't grabbed by simple mistakes.