Drag The Tiles To The Correct Boxes To Complete The Pairs.${ -6\left[\begin{array}{ccc} 3 & -1 & 3 \ 8 & X & -9 \ 4 & -1 & Y \end{array}\right]=\left[\begin{array}{ccc} -18 & U & -18 \ W & 6 & V \ -24 & 6 & 0 \end{array}\right] }$Match

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Introduction

Matrix equations are a fundamental concept in linear algebra, and solving them is a crucial skill for anyone working with matrices. In this article, we will explore how to solve matrix equations, focusing on the specific example of a 3x3 matrix equation. We will break down the solution step by step, using a combination of mathematical reasoning and matrix operations.

Understanding Matrix Equations

A matrix equation is an equation that involves matrices, where the matrices are combined using operations such as addition, subtraction, multiplication, and division. In this article, we will focus on solving a matrix equation of the form:

βˆ’6[3βˆ’138xβˆ’94βˆ’1y]=[βˆ’18uβˆ’18w6vβˆ’2460]{ -6\left[\begin{array}{ccc} 3 & -1 & 3 \\ 8 & x & -9 \\ 4 & -1 & y \end{array}\right]=\left[\begin{array}{ccc} -18 & u & -18 \\ w & 6 & v \\ -24 & 6 & 0 \end{array}\right] }

Step 1: Multiply the Matrix on the Left-Hand Side

To solve the matrix equation, we first need to multiply the matrix on the left-hand side by the scalar -6. This involves multiplying each element of the matrix by -6.

[βˆ’186βˆ’18βˆ’48βˆ’6x54βˆ’2460]{ \left[\begin{array}{ccc} -18 & 6 & -18 \\ -48 & -6x & 54 \\ -24 & 6 & 0 \end{array}\right] }

Step 2: Equate Corresponding Elements

Next, we need to equate corresponding elements of the two matrices. This involves setting up a system of linear equations, where each equation corresponds to a pair of corresponding elements.

{ \begin{align*} -18 &= -18 \\ -48 &= w \\ -24 &= -24 \\ 6 &= 6 \\ -6x &= 6 \\ 54 &= v \\ 0 &= 0 \end{align*} }

Step 3: Solve the System of Linear Equations

Now, we need to solve the system of linear equations. We can start by solving for x.

βˆ’6x=6{ -6x = 6 } x=βˆ’1{ x = -1 }

Step 4: Substitute the Value of x into the Matrix

Next, we need to substitute the value of x into the matrix.

[βˆ’186βˆ’18βˆ’48354βˆ’2460]{ \left[\begin{array}{ccc} -18 & 6 & -18 \\ -48 & 3 & 54 \\ -24 & 6 & 0 \end{array}\right] }

Step 5: Equate Corresponding Elements Again

Now, we need to equate corresponding elements of the two matrices again.

{ \begin{align*} -18 &= -18 \\ -48 &= w \\ -24 &= -24 \\ 6 &= 6 \\ 3 &= 6 \\ 54 &= v \\ 0 &= 0 \end{align*} }

Step 6: Solve for w and v

Now, we need to solve for w and v.

w=βˆ’48{ w = -48 } v=54{ v = 54 }

Step 7: Substitute the Values of w and v into the Matrix

Next, we need to substitute the values of w and v into the matrix.

[βˆ’186βˆ’18βˆ’48354βˆ’2460]{ \left[\begin{array}{ccc} -18 & 6 & -18 \\ -48 & 3 & 54 \\ -24 & 6 & 0 \end{array}\right] }

Conclusion

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

βˆ’6[3βˆ’138xβˆ’94βˆ’1y]=[βˆ’18uβˆ’18w6vβˆ’2460]{ -6\left[\begin{array}{ccc} 3 & -1 & 3 \\ 8 & x & -9 \\ 4 & -1 & y \end{array}\right]=\left[\begin{array}{ccc} -18 & u & -18 \\ w & 6 & v \\ -24 & 6 & 0 \end{array}\right] }

We have broken down the solution into several steps, using a combination of mathematical reasoning and matrix operations. We have solved for x, w, and v, and substituted the values into the matrix.

Tips and Tricks

  • When solving matrix equations, it is essential to follow the order of operations carefully.
  • When equating corresponding elements, make sure to set up a system of linear equations.
  • When solving the system of linear equations, use substitution or elimination methods.
  • When substituting values into the matrix, make sure to update the matrix correctly.

Common Mistakes

  • Failing to follow the order of operations carefully.
  • Failing to set up a system of linear equations when equating corresponding elements.
  • Failing to use substitution or elimination methods when solving the system of linear equations.
  • Failing to update the matrix correctly when substituting values.

Real-World Applications

Matrix equations have numerous real-world applications, including:

  • Computer graphics
  • Machine learning
  • Data analysis
  • Physics and engineering

Conclusion

Introduction

In our previous article, we explored how to solve matrix equations, focusing on the specific example of a 3x3 matrix equation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving matrix equations.

Q: What is a matrix equation?

A: A matrix equation is an equation that involves matrices, where the matrices are combined using operations such as addition, subtraction, multiplication, and division.

Q: How do I solve a matrix equation?

A: To solve a matrix equation, you need to follow these steps:

  1. Multiply the matrix on the left-hand side by the scalar.
  2. Equate corresponding elements of the two matrices.
  3. Set up a system of linear equations.
  4. Solve the system of linear equations using substitution or elimination methods.
  5. Substitute the values into the matrix.

Q: What is the order of operations when solving a matrix equation?

A: The order of operations when solving a matrix equation is:

  1. Multiply the matrix on the left-hand side by the scalar.
  2. Equate corresponding elements of the two matrices.
  3. Set up a system of linear equations.
  4. Solve the system of linear equations using substitution or elimination methods.
  5. Substitute the values into the matrix.

Q: How do I equate corresponding elements of two matrices?

A: To equate corresponding elements of two matrices, you need to set up a system of linear equations. For example, if you have two matrices:

[abcdefghi]{ \left[\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] }

and

[jklmnopqr]{ \left[\begin{array}{ccc} j & k & l \\ m & n & o \\ p & q & r \end{array}\right] }

you can equate corresponding elements by setting up the following system of linear equations:

{ \begin{align*} a &= j \\ b &= k \\ c &= l \\ d &= m \\ e &= n \\ f &= o \\ g &= p \\ h &= q \\ i &= r \end{align*} }

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use substitution or elimination methods. For example, if you have the following system of linear equations:

{ \begin{align*} 2x + 3y &= 7 \\ x - 2y &= -3 \end{align*} }

you can solve it using substitution or elimination methods.

Q: What is the difference between substitution and elimination methods?

A: Substitution and elimination methods are two different techniques used to solve systems of linear equations.

Substitution method:

  • Substitute one equation into the other equation.
  • Solve for the variable.

Elimination method:

  • Multiply one or both equations by a scalar.
  • Add or subtract the equations to eliminate one variable.
  • Solve for the remaining variable.

Q: How do I substitute values into a matrix?

A: To substitute values into a matrix, you need to update the matrix correctly. For example, if you have the following matrix:

[abcdefghi]{ \left[\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] }

and you want to substitute the values a = 2, b = 3, c = 4, d = 5, e = 6, f = 7, g = 8, h = 9, and i = 10, you can update the matrix as follows:

[2345678910]{ \left[\begin{array}{ccc} 2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 10 \end{array}\right] }

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving matrix equations. We have covered topics such as the order of operations, equating corresponding elements, solving systems of linear equations, and substituting values into a matrix. By following these steps and techniques, you can solve matrix equations with confidence.

Tips and Tricks

  • Make sure to follow the order of operations carefully.
  • Use substitution or elimination methods to solve systems of linear equations.
  • Update the matrix correctly when substituting values.
  • Practice solving matrix equations to become proficient.

Common Mistakes

  • Failing to follow the order of operations carefully.
  • Failing to use substitution or elimination methods to solve systems of linear equations.
  • Failing to update the matrix correctly when substituting values.
  • Failing to practice solving matrix equations.

Real-World Applications

Matrix equations have numerous real-world applications, including:

  • Computer graphics
  • Machine learning
  • Data analysis
  • Physics and engineering

Conclusion

In conclusion, solving matrix equations is a crucial skill for anyone working with matrices. By following the steps and techniques outlined in this article, you can solve matrix equations with confidence. Remember to practice solving matrix equations to become proficient, and don't hesitate to ask for help if you need it.