Consider The Matrix Equation Shown Below:$ \left[\begin{array}{c} 6x \\ 7y \\ -2z \end{array}\right] = \left[\begin{array}{c} 12 \\ 14 \\ 16 \end{array}\right] $Determine The Value Of Each

Introduction

In mathematics, matrix equations are a fundamental concept that plays a crucial role in various fields, including linear algebra, calculus, and engineering. A matrix equation is a mathematical statement that involves matrices and is used to represent a system of linear equations. In this article, we will focus on solving a matrix equation of the form:

$$\left[\begin{array}{c} 6x \\ 7y \\ -2z \end{array}\right] = \left[\begin{array}{c} 12 \\ 14 \\ 16 \end{array}\right]$$

Our goal is to determine the value of each variable $x$, $y$, and $z$ that satisfies the given matrix equation.

Understanding Matrix Equations

A matrix equation is a mathematical statement that involves matrices and is used to represent a system of linear equations. In the given matrix equation, we have a 3x1 matrix on the left-hand side and a 3x1 matrix on the right-hand side. The matrix on the left-hand side is a column matrix, also known as a vector, and the matrix on the right-hand side is also a column matrix.

To solve the matrix equation, we need to find the values of $x$, $y$, and $z$ that make the two matrices equal. This means that the corresponding elements of the two matrices must be equal.

Breaking Down the Matrix Equation

Let's break down the matrix equation into three separate equations, one for each element of the matrices:

$$6x = 12$$

$$7y = 14$$

$$-2z = 16$$

These three equations represent a system of linear equations, where each equation is a linear combination of the variables $x$, $y$, and $z$.

Solving the System of Linear Equations

To solve the system of linear equations, we can use various methods, including substitution, elimination, and matrices. In this case, we will use the substitution method to solve for each variable.

Solving for $x$

We can solve for $x$ by dividing both sides of the first equation by 6:

$$x = \frac{12}{6}$$

$$x = 2$$

Solving for $y$

We can solve for $y$ by dividing both sides of the second equation by 7:

$$y = \frac{14}{7}$$

$$y = 2$$

Solving for $z$

We can solve for $z$ by dividing both sides of the third equation by -2:

$$z = \frac{16}{-2}$$

$$z = -8$$

Conclusion

In this article, we have solved a matrix equation of the form:

$$\left[\begin{array}{c} 6x \\ 7y \\ -2z \end{array}\right] = \left[\begin{array}{c} 12 \\ 14 \\ 16 \end{array}\right]$$

We have broken down the matrix equation into three separate equations, one for each element of the matrices, and solved for each variable using the substitution method. The values of $x$, $y$, and $z$ that satisfy the given matrix equation are $x = 2$, $y = 2$, and $z = -8$.

Applications of Matrix Equations

Matrix equations have numerous applications in various fields, including:

• Linear Algebra: Matrix equations are used to represent systems of linear equations and to solve for the unknown variables.
• Calculus: Matrix equations are used to represent systems of differential equations and to solve for the unknown variables.
• Engineering: Matrix equations are used to represent systems of linear equations and to solve for the unknown variables in various engineering applications, such as circuit analysis and signal processing.
• Computer Science: Matrix equations are used to represent systems of linear equations and to solve for the unknown variables in various computer science applications, such as machine learning and data analysis.

Final Thoughts

Matrix equations are a fundamental concept in mathematics that plays a crucial role in various fields. In this article, we have solved a matrix equation of the form:

$$\left[\begin{array}{c} 6x \\ 7y \\ -2z \end{array}\right] = \left[\begin{array}{c} 12 \\ 14 \\ 16 \end{array}\right]$$

We have broken down the matrix equation into three separate equations, one for each element of the matrices, and solved for each variable using the substitution method. The values of $x$, $y$, and $z$ that satisfy the given matrix equation are $x = 2$, $y = 2$, and $z = -8$. Matrix equations have numerous applications in various fields, including linear algebra, calculus, engineering, and computer science.

Introduction

In our previous article, we solved a matrix equation of the form:

$$\left[\begin{array}{c} 6x \\ 7y \\ -2z \end{array}\right] = \left[\begin{array}{c} 12 \\ 14 \\ 16 \end{array}\right]$$

We broke down the matrix equation into three separate equations, one for each element of the matrices, and solved for each variable using the substitution method. In this article, we will answer some frequently asked questions about matrix equations.

Q: What is a matrix equation?

A: A matrix equation is a mathematical statement that involves matrices and is used to represent a system of linear equations.

Q: How do I solve a matrix equation?

A: To solve a matrix equation, you need to break down the matrix equation into separate equations, one for each element of the matrices, and then solve for each variable using a method such as substitution or elimination.

Q: What are the applications of matrix equations?

A: Matrix equations have numerous applications in various fields, including linear algebra, calculus, engineering, and computer science.

Q: Can I use a calculator to solve a matrix equation?

A: Yes, you can use a calculator to solve a matrix equation. Many calculators have built-in functions for solving systems of linear equations.

Q: How do I determine the order of a matrix?

A: The order of a matrix is determined by the number of rows and columns it has. For example, a 3x2 matrix has 3 rows and 2 columns.

Q: What is the difference between a matrix and a vector?

A: A matrix is a rectangular array of numbers, while a vector is a one-dimensional array of numbers.

Q: Can I use a matrix equation to solve a system of nonlinear equations?

A: No, matrix equations are used to solve systems of linear equations, not nonlinear equations.

Q: How do I represent a system of linear equations as a matrix equation?

A: To represent a system of linear equations as a matrix equation, you need to write the coefficients of the variables in a matrix and the constants in a column matrix.

Q: Can I use a matrix equation to solve a system of differential equations?

A: Yes, matrix equations can be used to solve systems of differential equations.

Q: How do I determine the solution to a matrix equation?

A: To determine the solution to a matrix equation, you need to solve for each variable using a method such as substitution or elimination.

Q: Can I use a matrix equation to solve a system of equations with more than one solution?

A: No, matrix equations are used to solve systems of linear equations, which have a unique solution.

Q: How do I represent a matrix equation in a computer program?

A: To represent a matrix equation in a computer program, you need to use a programming language such as MATLAB or Python.

Conclusion

In this article, we have answered some frequently asked questions about matrix equations. Matrix equations are a fundamental concept in mathematics that plays a crucial role in various fields. We hope that this article has provided you with a better understanding of matrix equations and how to solve them.

Final Thoughts

Matrix equations are a powerful tool for solving systems of linear equations. They have numerous applications in various fields, including linear algebra, calculus, engineering, and computer science. We hope that this article has provided you with a better understanding of matrix equations and how to solve them.

Additional Resources

For more information on matrix equations, we recommend the following resources:

• Linear Algebra: A textbook on linear algebra that covers matrix equations in detail.
• Calculus: A textbook on calculus that covers matrix equations in detail.
• Engineering: A textbook on engineering that covers matrix equations in detail.
• Computer Science: A textbook on computer science that covers matrix equations in detail.

We hope that this article has provided you with a better understanding of matrix equations and how to solve them. If you have any further questions, please don't hesitate to ask.