If ∣ 2 X + 3 2 X − 1 4 ∣ = 2 \left|\begin{array}{cc}2x+3 & 2 \\ X-1 & 4\end{array}\right|=2 ​ 2 X + 3 X − 1 ​ 2 4 ​ ​ = 2 , Then Find The Value Of X X X .

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Introduction

Determinants are a crucial concept in linear algebra, and they have numerous applications in various fields, including mathematics, physics, and engineering. In this article, we will explore the concept of determinants and how to find the value of a variable using a determinant equation. Specifically, we will solve the determinant equation $\left|\begin{array}{cc}2x+3 & 2 \\ x-1 & 4\end{array}\right|=2$ to find the value of $x$.

What is a Determinant?

A determinant is a scalar value that can be calculated from the elements of a square matrix. It is denoted by the symbol $\det(A)$ or $|A|$. The determinant of a 2x2 matrix $\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]$ is calculated as follows:

$$\det(A) = ad - bc$$

Calculating the Determinant

To calculate the determinant of the given matrix, we will use the formula:

$$\det(A) = (2x+3)(4) - (2)(x-1)$$

Expanding the equation, we get:

$$\det(A) = 8x + 12 - 2x + 2$$

Simplifying the equation, we get:

$$\det(A) = 6x + 14$$

Setting Up the Equation

We are given that the determinant is equal to 2, so we can set up the equation:

$$6x + 14 = 2$$

Solving for $x$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 14 from both sides of the equation:

$$6x = -12$$

Next, we can divide both sides of the equation by 6 to solve for $x$:

$$x = -2$$

Conclusion

In this article, we have explored the concept of determinants and how to find the value of a variable using a determinant equation. We have solved the determinant equation $\left|\begin{array}{cc}2x+3 & 2 \\ x-1 & 4\end{array}\right|=2$ to find the value of $x$. The final answer is $x = -2$.

Final Answer

The final answer is $\boxed{-2}$.

Related Topics

• Determinants of 2x2 matrices
• Solving linear equations using determinants
• Applications of determinants in mathematics and physics

References

• Linear Algebra and Its Applications by Gilbert Strang
• Determinants and Matrices by Math Is Fun

Further Reading

• Determinants of 3x3 matrices
• Solving systems of linear equations using determinants
  

Introduction

Determinants are a fundamental concept in linear algebra, and they have numerous applications in various fields, including mathematics, physics, and engineering. In this article, we will explore the concept of determinants and answer some frequently asked questions about determinants and linear algebra.

Q: What is a determinant?

A: A determinant is a scalar value that can be calculated from the elements of a square matrix. It is denoted by the symbol $\det(A)$ or $|A|$. The determinant of a 2x2 matrix $\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]$ is calculated as follows:

$$\det(A) = ad - bc$$

Q: What is the purpose of determinants?

A: Determinants have several purposes in linear algebra. They can be used to:

• Determine the invertibility of a matrix
• Find the solution to a system of linear equations
• Calculate the area or volume of a parallelogram or a parallelepiped
• Solve linear systems using Cramer's rule

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

A: The determinant of a 3x3 matrix $\left[\begin{array}{ccc}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ is calculated as follows:

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

Q: What is the difference between a determinant and an adjugate?

A: The determinant and the adjugate are two related but distinct concepts in linear algebra. The determinant is a scalar value that can be calculated from the elements of a square matrix, while the adjugate is a matrix that is obtained by replacing each element of the original matrix with its cofactor.

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

A: Determinants can be used to solve a system of linear equations using Cramer's rule. This involves replacing each element of the original matrix with its corresponding cofactor and then calculating the determinant of the resulting matrix.

Q: What is the relationship between determinants and eigenvalues?

A: Determinants and eigenvalues are related in that the determinant of a matrix is equal to the product of its eigenvalues. This means that if a matrix has a determinant of zero, then it must have at least one eigenvalue of zero.

Q: Can you provide some examples of how determinants are used in real-world applications?

A: Yes, determinants have numerous applications in various fields, including:

• Physics: Determinants are used to calculate the area or volume of a parallelogram or a parallelepiped.
• Engineering: Determinants are used to solve systems of linear equations and to calculate the stress and strain on a structure.
• Computer Science: Determinants are used in algorithms for solving systems of linear equations and in computer graphics.

Q: What are some common mistakes to avoid when working with determinants?

A: Some common mistakes to avoid when working with determinants include:

• Failing to check for zero determinant before inverting a matrix
• Failing to calculate the determinant correctly
• Failing to use the correct formula for the determinant of a matrix

Conclusion

In this article, we have explored the concept of determinants and answered some frequently asked questions about determinants and linear algebra. We hope that this article has provided a helpful introduction to the topic and has clarified any confusion you may have had.

Final Answer

The final answer is $\boxed{0}$.

Related Topics

• Determinants of 2x2 matrices
• Determinants of 3x3 matrices
• Solving systems of linear equations using determinants
• Applications of determinants in mathematics and physics

References

• Linear Algebra and Its Applications by Gilbert Strang
• Determinants and Matrices by Math Is Fun

Further Reading

• Determinants of 4x4 matrices
• Solving systems of linear equations using Cramer's rule