Find Values For The Variables So That The Following Matrices Are Equal.$\[ \begin{bmatrix} x & 6y \\ z & 6 \end{bmatrix} = \begin{bmatrix} 18 & 6 \\ 8 & 6 \end{bmatrix} \\]$\[ X = \square \\] (Simplify Your Answer.) $\[ Y =

Introduction

In mathematics, matrices are a fundamental concept used to represent systems of linear equations, linear transformations, and other mathematical structures. When two matrices are said to be equal, it means that they have the same elements in the same positions. In this article, we will explore how to find values for the variables so that the following matrices are equal.

The Given Matrices

We are given two matrices:

$$\begin{bmatrix} x & 6y \\ z & 6 \end{bmatrix} = \begin{bmatrix} 18 & 6 \\ 8 & 6 \end{bmatrix}$$

Our goal is to find the values of $x$, $y$, and $z$ that make these two matrices equal.

Equating Corresponding Elements

To find the values of $x$, $y$, and $z$, we need to equate the corresponding elements of the two matrices. This means that we need to set up a system of equations based on the equality of the matrices.

Looking at the first row of the matrices, we can see that:

$$x = 18$$

This is because the element in the first row and first column of the first matrix is $x$, and the element in the first row and first column of the second matrix is $18$. Therefore, we can conclude that $x = 18$.

Solving for y

Now that we have found the value of $x$, we can move on to solving for $y$. Looking at the first row of the matrices, we can see that:

$$6y = 6$$

This is because the element in the first row and second column of the first matrix is $6y$, and the element in the first row and second column of the second matrix is $6$. To solve for $y$, we can divide both sides of the equation by $6$:

$$y = \frac{6}{6}$$

$$y = 1$$

Solving for z

Now that we have found the values of $x$ and $y$, we can move on to solving for $z$. Looking at the second row of the matrices, we can see that:

$$z = 8$$

This is because the element in the second row and first column of the first matrix is $z$, and the element in the second row and first column of the second matrix is $8$. Therefore, we can conclude that $z = 8$.

Conclusion

In conclusion, we have found the values of $x$, $y$, and $z$ that make the two matrices equal. The values are:

$$x = 18$$

$$y = 1$$

$$z = 8$$

These values satisfy the equality of the two matrices, and we can use them to solve systems of linear equations or other mathematical problems.

Final Answer

The final answer is:

• $x = 18$
• $y = 1$
• $z = 8$
  

Introduction

In our previous article, we explored how to find values for the variables so that the following matrices are equal. We set up a system of equations based on the equality of the matrices and solved for the values of $x$, $y$, and $z$. In this article, we will answer some common questions related to this topic.

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

A: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A vector is a one-dimensional array of numbers, symbols, or expressions. In other words, a matrix has multiple rows and columns, while a vector has only one row or column.

Q: How do I know if two matrices are equal?

A: Two matrices are equal if and only if they have the same elements in the same positions. This means that if you have two matrices, you can compare them element by element to see if they are equal.

Q: What is the purpose of matrices in mathematics?

A: Matrices are used to represent systems of linear equations, linear transformations, and other mathematical structures. They are also used in computer science, physics, engineering, and other fields to solve problems and model real-world situations.

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

A: To solve a system of linear equations using matrices, you can represent the system as a matrix equation and then use techniques such as Gaussian elimination or matrix inversion to solve for the variables.

Q: What is the difference between a square matrix and a non-square matrix?

A: A square matrix is a matrix that has the same number of rows and columns. A non-square matrix is a matrix that has a different number of rows and columns.

Q: How do I find the inverse of a matrix?

A: To find the inverse of a matrix, you can use techniques such as Gaussian elimination or matrix inversion. The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.

Q: What is the identity matrix?

A: The identity matrix is a square matrix that has 1s on the main diagonal and 0s elsewhere. It is used as a multiplicative identity in matrix multiplication.

Q: How do I add or subtract matrices?

A: To add or subtract matrices, you can add or subtract the corresponding elements of the matrices. This means that if you have two matrices, you can add or subtract them element by element.

Q: What is the transpose of a matrix?

A: The transpose of a matrix is a matrix that is obtained by interchanging the rows and columns of the original matrix. It is denoted by the symbol $A^T$.

Q: How do I find the determinant of a matrix?

A: To find the determinant of a matrix, you can use techniques such as expansion by minors or cofactor expansion. The determinant of a matrix is a scalar value that can be used to determine the invertibility of the matrix.

Q: What is the purpose of determinants in mathematics?

A: Determinants are used to determine the invertibility of a matrix, as well as to solve systems of linear equations and other mathematical problems.

Conclusion

In conclusion, we have answered some common questions related to finding values for the variables so that the following matrices are equal. We have discussed the difference between a matrix and a vector, how to know if two matrices are equal, and the purpose of matrices in mathematics. We have also discussed how to solve a system of linear equations using matrices, how to find the inverse of a matrix, and how to add or subtract matrices.

Final Answer

The final answer is:

• $x = 18$
• $y = 1$
• $z = 8$

We hope that this article has been helpful in answering your questions and providing a better understanding of matrices and linear algebra.