Problem 35 - Algebra and Trigonometry

Question

Write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rrr} {2} & {0} & {-1} \\ {0} & {3} & {0} \\ {1} & {1} & {0} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \\ {z} \end{array}\right]=\left[\begin{array}{l} {6} \\ {9} \\ {5} \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations is: \[2x - z = 6\], \[3y = 9\], and \[x + y = 5\].

Step 1: Multiply the First Row by the Column Vector

Multiply each element of the first row of the matrix by the corresponding element of the vector, and sum the results. This gives the first equation. So: \(2x + 0*y - 1z = 6\) which simplifies to \(2x - z = 6\).

Step 2: Multiply the Second Row by the Column Vector

Similarly, multiply each element of the second row by the corresponding element of the vector. This gives the second equation. So: \(0*x + 3y + 0*z = 9\) which simplifies to \(3y = 9\).

Step 3: Multiply the Third Row by the Column Vector

Finally, multiply each element of the third row by the corresponding element of the vector. This gives the third equation. So: \(1x + 1*y + 0*z = 5\) which simplifies to \(x + y = 5\).

Key Concepts

Linear Algebra

Linear algebra is an area of mathematics focused on vectors, spaces, and linear mappings between these spaces. It is a key foundation for many areas of mathematics and underpins a wide range of scientific and engineering disciplines.

In the context of solving matrix equations, linear algebra provides the framework and methods for finding the values of variables that satisfy a set of linear relationships. These variables are often represented by vectors while the relationships between them are captured using matrices. By applying linear algebra techniques, we can translate complex problems into systems that are more approachable and solvable.

System of Linear Equations

A system of linear equations is a collection of one or more equations involving the same variables where each equation is linear. This means that in each equation, the variables appear to the first power and multiply by constants. In the provided exercise, the system in question comprises three linear equations.

When represented without matrices, each equation corresponds to a line on a graph, and the solution to the system is the point or points where the lines intersect. To solve these systems, various methods can be used, such as substitution, elimination, and matrix operations. The goal is to isolate each variable and find their values, which fit all the given equations simultaneously.

Matrix Multiplication

Matrix multiplication is an operation that produces a new matrix from two matrices by multiplying corresponding entries and summing them up for each position in the resulting matrix. It's an essential procedure in linear algebra because it allows us to compactly represent and solve systems of linear equations.

In the provided exercise, we multiply a 3x3 matrix by a 3x1 column vector. Here's how it works in a generalized form: If we have a matrix \(A\) and a vector \(v\), their multiplication \(A * v\) is done by taking the dot product of each row of \(A\) with \(v\), which gives us a new column vector. This vector contains the solutions to our system of linear equations, as shown in the step-by-step solution.