Solve The System Using A Matrix:${ \begin{array}{c} \left{ \begin{array}{c} -x + Y - 3z = -4 \ 3x - 2y + 8z = 14 \ 2x - 2y + 5z = 7 \end{array} \right. \end{array} }$Give Your Answer As An Ordered Triple (x, Y, Z).

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Introduction

In this article, we will explore the method of solving a system of linear equations using a matrix. This method is a powerful tool for solving systems of linear equations and is widely used in various fields such as physics, engineering, and computer science. We will use the given system of linear equations to demonstrate the steps involved in solving a system using a matrix.

The System of Linear Equations

The given system of linear equations is:

${ \begin{array}{c} \left\{ \begin{array}{c} -x + y - 3z = -4 \\ 3x - 2y + 8z = 14 \\ 2x - 2y + 5z = 7 \end{array} \right. \end{array} \}$

Converting the System to a Matrix Equation

To solve the system using a matrix, we need to convert it into a matrix equation. We can do this by representing the coefficients of the variables as a matrix, the variables as a column matrix, and the constants as a column matrix.

Let's represent the coefficients of the variables as a matrix A, the variables as a column matrix X, and the constants as a column matrix B.

${ \begin{array}{c} A = \begin{bmatrix} -1 & 1 & -3 \\ 3 & -2 & 8 \\ 2 & -2 & 5 \end{bmatrix} \\ X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \\ B = \begin{bmatrix} -4 \\ 14 \\ 7 \end{bmatrix} \end{array} \}$

Finding the Inverse of Matrix A

To solve the system using a matrix, we need to find the inverse of matrix A. The inverse of a matrix A is denoted by A^(-1) and is calculated using the following formula:

${ A^{-1} = \frac{1}{det(A)} adj(A) }$

where det(A) is the determinant of matrix A and adj(A) is the adjoint of matrix A.

The determinant of matrix A is calculated as follows:

${ det(A) = (-1)^{1+1} \begin{vmatrix} -2 & 8 \\ -2 & 5 \end{vmatrix} + (-1)^{1+2} \begin{vmatrix} 3 & 8 \\ 2 & 5 \end{vmatrix} + (-1)^{1+3} \begin{vmatrix} 3 & -2 \\ 2 & -2 \end{vmatrix} }$

Simplifying the above expression, we get:

${ det(A) = (-2)(5) - (8)(-2) + (3)(-2) - (8)(2) + (3)(-2) - (2)(-2) }$

${ det(A) = -10 + 16 - 6 - 16 - 6 + 4 }$

${ det(A) = -18 }$

The adjoint of matrix A is calculated as follows:

${ adj(A) = \begin{bmatrix} -2 & 8 & 6 \\ -2 & 5 & -4 \\ 3 & -2 & 2 \end{bmatrix} }$

Now, we can calculate the inverse of matrix A as follows:

${ A^{-1} = \frac{1}{-18} \begin{bmatrix} -2 & 8 & 6 \\ -2 & 5 & -4 \\ 3 & -2 & 2 \end{bmatrix} }$

${ A^{-1} = \begin{bmatrix} \frac{1}{9} & -\frac{4}{9} & -\frac{1}{3} \\ \frac{1}{9} & -\frac{5}{18} & \frac{2}{9} \\ -\frac{1}{6} & \frac{1}{9} & -\frac{1}{9} \end{bmatrix} }$

Solving the System using the Inverse of Matrix A

Now that we have the inverse of matrix A, we can solve the system using the following formula:

${ X = A^{-1} B }$

Substituting the values of A^(-1) and B, we get:

${ X = \begin{bmatrix} \frac{1}{9} & -\frac{4}{9} & -\frac{1}{3} \\ \frac{1}{9} & -\frac{5}{18} & \frac{2}{9} \\ -\frac{1}{6} & \frac{1}{9} & -\frac{1}{9} \end{bmatrix} \begin{bmatrix} -4 \\ 14 \\ 7 \end{bmatrix} }$

Multiplying the matrices, we get:

${ X = \begin{bmatrix} \frac{1}{9}(-4) - \frac{4}{9}(14) - \frac{1}{3}(7) \\ \frac{1}{9}(-4) - \frac{5}{18}(14) + \frac{2}{9}(7) \\ -\frac{1}{6}(-4) + \frac{1}{9}(14) - \frac{1}{9}(7) \end{bmatrix} }$

Simplifying the above expression, we get:

${ X = \begin{bmatrix} -\frac{4}{9} - \frac{56}{9} - \frac{7}{3} \\ -\frac{4}{9} - \frac{70}{18} + \frac{14}{9} \\ \frac{2}{3} + \frac{14}{9} - \frac{7}{9} \end{bmatrix} }$

${ X = \begin{bmatrix} -\frac{4}{9} - \frac{56}{9} - \frac{21}{9} \\ -\frac{4}{9} - \frac{35}{9} + \frac{14}{9} \\ \frac{6}{9} + \frac{14}{9} - \frac{7}{9} \end{bmatrix} }$

${ X = \begin{bmatrix} -\frac{81}{9} \\ -\frac{25}{9} \\ \frac{13}{9} \end{bmatrix} }$

${ X = \begin{bmatrix} -9 \\ -\frac{25}{9} \\ \frac{13}{9} \end{bmatrix} }$

Conclusion

In this article, we have demonstrated the method of solving a system of linear equations using a matrix. We have used the given system of linear equations to demonstrate the steps involved in solving a system using a matrix. We have found

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Q: What is the main advantage of using a matrix to solve a system of linear equations?

A: The main advantage of using a matrix to solve a system of linear equations is that it provides a systematic and efficient way of solving the system. It also allows us to use the properties of matrices, such as the inverse of a matrix, to solve the system.

Q: How do I know if a matrix has an inverse?

A: A matrix has an inverse if and only if its determinant is non-zero. If the determinant is zero, then the matrix does not have an inverse.

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

A: To find the inverse of a matrix, you need to calculate the determinant of the matrix and then use the formula for the inverse of a matrix, which is given by:

${ A^{-1} = \frac{1}{det(A)} adj(A) }$

where det(A) is the determinant of the matrix and adj(A) is the adjoint of the matrix.

Q: What is the adjoint of a matrix?

A: The adjoint of a matrix is a matrix that is obtained by taking the transpose of the matrix and then replacing each element with its cofactor.

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

A: To calculate the determinant of a matrix, you need to use the formula for the determinant of a 3x3 matrix, which is given by:

${ det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) }$

where a, b, c, d, e, f, g, h, and i are the elements of the matrix.

Q: What is the difference between the inverse of a matrix and the adjoint of a matrix?

A: The inverse of a matrix is a matrix that, when multiplied by the original matrix, gives the identity matrix. The adjoint of a matrix is a matrix that is used to calculate the inverse of the matrix.

Q: Can I use a matrix to solve a system of linear equations with more than three variables?

A: Yes, you can use a matrix to solve a system of linear equations with more than three variables. However, the matrix will be larger and more complex, and you will need to use more advanced techniques to solve the system.

Q: Are there any limitations to using a matrix to solve a system of linear equations?

A: Yes, there are several limitations to using a matrix to solve a system of linear equations. For example, the matrix must have an inverse, and the system must be consistent. Additionally, the matrix may be too large or too complex to solve using a matrix.

Q: Can I use a matrix to solve a system of linear equations with complex coefficients?

A: Yes, you can use a matrix to solve a system of linear equations with complex coefficients. However, you will need to use complex numbers and complex arithmetic to solve the system.

Q: Are there any other methods for solving a system of linear equations besides using a matrix?

A: Yes, there are several other methods for solving a system of linear equations, including substitution, elimination, and graphing. However, using a matrix is often the most efficient and systematic way to solve a system of linear equations.

Q: Can I use a matrix to solve a system of linear equations with inequalities?

A: No, you cannot use a matrix to solve a system of linear equations with inequalities. Inequalities require a different approach, such as using linear programming or graphing.

Q: Are there any software packages or tools that can help me solve a system of linear equations using a matrix?

A: Yes, there are several software packages and tools that can help you solve a system of linear equations using a matrix, including MATLAB, Mathematica, and Python.