Problem 36 - Algebra and Trigonometry

Question

Write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rrr} {-1} & {0} & {1} \\ {0} & {-1} & {0} \\ {0} & {1} & {1} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \\ {z} \end{array}\right]=\left[\begin{array}{r} {-4} \\ {2} \\ {4} \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations corresponding to the matrix equation are: \( -x + z = -4 \), \( -y=2 \), and \( y + z = 4 \).

Step 1: Formulate the first linear equation

The first row of the left matrix is \( -1, 0, 1 \). This can be re-written as \(-1*x + 0*y + 1*z\). Given the right manually, the equation would be \( -1*x + 0*y + 1*z = -4 \) or simplified as \( -x + z = -4 \).

Step 2: Formulate the second linear equation

The second row of the left matrix is \(0,-1,0\). This can be translated to \(0*x - 1*y + 0*z\). The second element on the right matrix is \(2\), making the equation \(0*x - 1*y + 0*z = 2 \) or simplified as \( -y=2 \).

Step 3: Formulate the third linear equation

The third row of the left matrix is \(0,1,1\). This can be translated to \(0*x + 1*y + 1*z\). The third element on the right matrix is \(4\), making the equation \(0*x + 1*y + 1*z = 4 \) or simplified as \( y + z = 4 \).

Key Concepts

Linear Equations

Linear equations are foundational to understanding algebra and many higher-level math concepts. A linear equation is an equation that forms a straight line when graphed. The general form of a linear equation in two variables is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. In more advanced applications, like in matrices, variables can extend to more than two, as seen in the exercise.Each matrix row corresponds to a linear equation where each element in the row is a coefficient of variables. For example, in the first row \( -1x + 0y + 1z = -4 \) represents the relationship among \( x \), \( y \), and \( z \). Notice how the matrix entry directly translates into equation coefficients.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Solving a system of equations means finding the values of the unknowns that satisfy all the equations simultaneously.In the exercise, converting the matrix equation into a system of equations yields:

• \( -x + z = -4 \)
• \( -y = 2 \)
• \( y + z = 4 \)

These three equations represent a system that can be solved using techniques like substitution or elimination. Analyzing the equations helps to determine the values of \( x \), \( y \), and \( z \) that satisfy all three equations. Just like working a puzzle, each equation provides a piece of information needed to find the solution.

Matrix Operations

Matrix operations are mathematical procedures that involve the manipulation of matrices, which are rectangular arrays of numbers used to represent data or equations. Common operations include addition, subtraction, and multiplication. In the realm of linear equations, matrix multiplication is a key operation. When a matrix is multiplied by a vector of variables (like in the exercise), the resulting vector provides the constant values for the system of equations they represent. Each element from the resultant vector comes from the dot product of a row of the matrix and the column matrix (or vector) of variables. The provided exercise illustrates this perfectly, turning matrix multiplication into the system of linear equations. Understanding these operations is critical because they allow us to solve complex systems efficiently using matrix algebra methods.