Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{aligned} x-3 y+z &=1 \\ 4 y &=0 \\ 7 z &=-5 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The augmented matrix for the given system of equations is \[ \left[ \begin{array}{ccc|c} 1 & -3 & 1 & 1 \ 0 & 4 & 0 & 0 \ 0 & 0 & 7 & -5 \end{array} \right] \] and its dimension is 3 x 4.

Step 1: Understand the Augmented Matrix

An augmented matrix is derived from a system of equations, with each row representing a single equation, and each column represents an individual variable, and the final column representing the constant term from each equation. So, an augmented matrix will have the form \[ [a_1, b_1, ..., z_1 | w_1], [a_2, b_2, ..., z_2 | w_2], ..., [a_n, b_n, ..., z_n | w_n]] \], where a, b, ..., z are coefficients of the variables and w is the constant on the right side of the equation. The '|' symbol shows the division between the coefficients on the left and the constants on the right.

Step 2: Convert Given Equations to Augmented Matrix

From the given equations, we write the augmented matrix \[ \left[ \begin{array}{ccc|c} 1 & -3 & 1 & 1 \ 0 & 4 & 0 & 0 \ 0 & 0 & 7 & -5 \end{array} \right] \] The coefficients for each variable in every equation are put into the matrix, and the constants on the right side of the equal sign are placed in the final column after the '|'. Here, the absence of a variable in an equation is represented as 0 in the respective position in the matrix.

Step 3: Identify the Matrix Dimension

The dimension of a matrix is defined by the number of rows and columns it has. Here, our matrix has 3 rows and 4 columns (including the augmented column). So this is a 3 x 4 dimension matrix.

Key Concepts

system of linear equations

When you have more than one linear equation working together, you're dealing with a system of linear equations. In these systems:

• Each equation appears as a straight line when graphed.
• The solution is where all the lines meet at a single point, meaning they all share the same x, y, and possibly z values.

Systems of linear equations often represent complex real-world scenarios, like finding the intersection of different budget constraints. In math, solving these systems typically involves expressing equations in forms that make comparisons and manipulations easier, such as matrices.

matrix dimension

The term 'matrix dimension' refers to the size of a matrix, which is expressed in terms of rows and columns. For our particular augmented matrix, we determine its dimension by:

• Counting the number of rows. Each row corresponds to an equation.
• Counting the number of columns, including the column for constants on the right side of the '|'.

So for our matrix, which represents three equations, it has three rows. It also includes an extra column for constants, totaling to four columns altogether. Thus, this matrix is called a 3 x 4 matrix, succinctly indicating its structure.

linear algebra

Linear algebra is a fundamental branch of mathematics focusing on vectors, matrices, and their transformations. It presents countless ways to solve systems of equations, such as through matrix operations.
Within linear algebra:

• Matrices help organize coefficients and constants of equations neatly.
• Operations with matrices, like row manipulations, can simplify solving systems.
• The concepts of span, linear combinations, and transformations aid in understanding geometric interpretations of these systems.

The power of linear algebra lies in its ability to handle vast networks of equations, making it invaluable not only in mathematics but also in engineering, computer science, and economics.