What Is The Determinant Of The Coefficient Matrix Of The System $\[ \begin{cases} 4x - 6y + 0z = -7 \\ 3x + 3y + 0z = -2 \\ 2x - 12y + 0z = -1 \end{cases} \\]A. $-7$ B. $-2$ C. $-1$ D. 0

Introduction

In linear algebra, the coefficient matrix of a system of linear equations is a matrix that contains the coefficients of the variables in the equations. The determinant of this matrix is an important concept in solving systems of linear equations, as it can be used to determine the existence and uniqueness of solutions. In this article, we will discuss the determinant of the coefficient matrix of a given system of linear equations and provide a step-by-step solution to find the determinant.

The System of Linear Equations

The given system of linear equations is:

$$\begin{cases} 4x - 6y + 0z = -7 \\ 3x + 3y + 0z = -2 \\ 2x - 12y + 0z = -1 \end{cases}$$

The Coefficient Matrix

The coefficient matrix of the system is:

$$A = \begin{bmatrix} 4 & -6 & 0 \\ 3 & 3 & 0 \\ 2 & -12 & 0 \end{bmatrix}$$

Finding the Determinant

To find the determinant of the coefficient matrix, we can use the formula for the determinant of a 3x3 matrix:

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

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

Step 1: Identify the Elements of the Matrix

The elements of the matrix are:

$a = 4$, $b = -6$, $c = 0$, $d = 3$, $e = 3$, $f = 0$, $g = 2$, $h = -12$, and $i = 0$

Step 2: Plug the Elements into the Formula

Plugging the elements into the formula, we get:

$$\det(A) = 4(3(0) - 0(-12)) - (-6)(3(0) - 0(2)) + 0(3(-12) - 2(3))$$

Step 3: Simplify the Expression

Simplifying the expression, we get:

$$\det(A) = 4(0 - 0) + 6(0 - 0) + 0(-36 - 6)$$

Step 4: Evaluate the Expression

Evaluating the expression, we get:

$$\det(A) = 4(0) + 6(0) + 0(-42)$$

Step 5: Simplify the Expression

Simplifying the expression, we get:

$$\det(A) = 0 + 0 + 0$$

Step 6: Evaluate the Expression

Evaluating the expression, we get:

$$\det(A) = 0$$

Conclusion

The determinant of the coefficient matrix of the system is 0. This means that the system has either no solution or infinitely many solutions.

Discussion

The determinant of the coefficient matrix is an important concept in solving systems of linear equations. It can be used to determine the existence and uniqueness of solutions. In this case, the determinant is 0, which means that the system has either no solution or infinitely many solutions.

Final Answer

The final answer is 0.

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Introduction to Linear Algebra, 5th Edition, by Gilbert Strang
• [3] Linear Algebra, 2nd Edition, by David C. Lay

Related Topics

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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• Mathematics
• Linear Algebra

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• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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The determinant of the coefficient matrix of a system of linear equations is an important concept in solving systems of linear equations. In this article, we will discuss the determinant of the coefficient matrix of a given system of linear equations and provide a step-by-step solution to find the determinant.

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• Coefficient Matrix
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What is the Determinant of the Coefficient Matrix of the System?

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The determinant of the coefficient matrix of a system of linear equations is an important concept in solving systems of linear equations. In this article, we will discuss the determinant of the coefficient matrix of a given system of linear equations and provide a step-by-step solution to find the determinant.

Related Articles

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Related Topics

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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• Mathematics
• Linear Algebra

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• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

SEO Description

The determinant of the coefficient matrix of a system of linear equations is an important concept in solving systems of linear equations. In this article, we will discuss the determinant of the coefficient matrix of a given system of linear equations and provide a step-by-step solution to find the determinant.

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• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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What is the Determinant of the Coefficient Matrix of the System?

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The determinant of the coefficient matrix of a system of linear equations is an important concept in solving systems of linear equations. In this article, we will discuss the determinant of the coefficient matrix of a given system of linear equations and provide a step-by-step solution to find the determinant.

Related Articles

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Related Topics

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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• Mathematics
• Linear Algebra

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• Determinant
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

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• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra
  

Introduction

Determinants of coefficient matrices are an essential concept in linear algebra, and understanding them can help you solve systems of linear equations more efficiently. In this article, we will answer some frequently asked questions about determinants of coefficient matrices.

Q: What is the determinant of a matrix?

A: The determinant of a matrix is a scalar value that can be calculated from the elements of the matrix. It is used to determine the solvability of a system of linear equations and the rank of a matrix.

Q: How do I calculate the determinant of a 3x3 matrix?

A: To calculate the determinant of a 3x3 matrix, you can use the formula:

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

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

Q: What is the significance of the determinant of a coefficient matrix?

A: The determinant of a coefficient matrix is used to determine the solvability of a system of linear equations. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system has either no solution or infinitely many solutions.

Q: How do I determine the solvability of a system of linear equations?

A: To determine the solvability of a system of linear equations, you can calculate the determinant of the coefficient matrix. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system has either no solution or infinitely many solutions.

Q: What is the relationship between the determinant of a matrix and its inverse?

A: The determinant of a matrix is used to calculate its inverse. If the determinant of a matrix is non-zero, the matrix is invertible, and its inverse can be calculated using the formula:

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} e & f & g \\ h & i & j \\ k & l & m \end{bmatrix}$$

where $e$, $f$, $g$, $h$, $i$, $j$, $k$, $l$, and $m$ are the elements of the matrix.

Q: Can I use the determinant of a matrix to determine its rank?

A: Yes, you can use the determinant of a matrix to determine its rank. If the determinant of a matrix is non-zero, the matrix is of full rank, and its rank is equal to the number of rows or columns. If the determinant of a matrix is zero, the matrix is not of full rank, and its rank is less than the number of rows or columns.

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

A: To calculate the determinant of a matrix using a calculator, you can enter the matrix into the calculator and use the "det" or "determinant" function. The calculator will then calculate the determinant of the matrix and display the result.

Q: Can I use the determinant of a matrix to solve a system of linear equations?

A: Yes, you can use the determinant of a matrix to solve a system of linear equations. If the determinant of the coefficient matrix is non-zero, you can use the formula:

$$x = \frac{\det(Ab)}{\det(A)}$$

where $A$ is the coefficient matrix, $b$ is the constant matrix, and $x$ is the solution vector.

Q: What is the relationship between the determinant of a matrix and its eigenvalues?

A: The determinant of a matrix is equal to the product of its eigenvalues. If the determinant of a matrix is non-zero, the matrix has a non-zero eigenvalue. If the determinant of a matrix is zero, the matrix has a zero eigenvalue.

Q: Can I use the determinant of a matrix to determine its orthogonality?

A: Yes, you can use the determinant of a matrix to determine its orthogonality. If the determinant of a matrix is non-zero, the matrix is orthogonal. If the determinant of a matrix is zero, the matrix is not orthogonal.

Conclusion

Determinants of coefficient matrices are an essential concept in linear algebra, and understanding them can help you solve systems of linear equations more efficiently. In this article, we have answered some frequently asked questions about determinants of coefficient matrices.

Final Answer

The final answer is that the determinant of a coefficient matrix is a scalar value that can be calculated from the elements of the matrix. It is used to determine the solvability of a system of linear equations and the rank of a matrix.

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Introduction to Linear Algebra, 5th Edition, by Gilbert Strang
• [3] Linear Algebra, 2nd Edition, by David C. Lay

Related Topics

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Tags

• Determinant
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Keywords

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

SEO Description

Determinants of coefficient matrices are an essential concept in linear algebra, and understanding them can help you solve systems of linear equations more efficiently. In this article, we have answered some frequently asked questions about determinants of coefficient matrices.

SEO Keywords

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

SEO Title

Frequently Asked Questions (FAQs) About Determinants of Coefficient Matrices

SEO Description

Determinants of coefficient matrices are an essential concept in linear algebra, and understanding them can help you solve systems of linear equations more efficiently. In this article, we have answered some frequently asked questions about determinants of coefficient matrices.

Related Articles

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Related Topics

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Categories

• Mathematics
• Linear Algebra

Tags

• Determinant
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra

Keywords

• Determinant of a Matrix
• Coefficient Matrix
• System of Linear Equations
• Linear Algebra