If ∣ 2 X 5 12 X ∣ = ∣ 6 − 5 4 3 ∣ \left|\begin{array}{ll}2 X & 5 \\ 12 & X \end{array}\right| = \left|\begin{array}{rr}6 & -5 \\ 4 & 3\end{array}\right| 2 X 12 5 X = 6 4 − 5 3 , Then The Value Of X X X Is:(A) 3 3 3 (B) 7 7 7 (C) ± 7 \pm 7 ± 7 (D) $\pm

Mar 9, 2025 by ADMIN 268 views

**Determinant of a 2x2 Matrix: A Comprehensive Guide**

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Introduction

In mathematics, a determinant is a scalar value that can be calculated from the elements of a square matrix. It is a fundamental concept in linear algebra and has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will explore the concept of determinants of a 2x2 matrix and use it to solve a problem involving absolute values.

What is a Determinant?

A determinant of a 2x2 matrix is calculated using the following formula:

\left|\begin{array}{ll}a & b \\ c & d\end{array}\right| = ad - bc

where $a$ , $b$ , $c$ , and $d$ are the elements of the matrix.

Properties of Determinants

Determinants have several important properties that make them useful in various applications. Some of the key properties include:

Multiplication property: The determinant of a product of two matrices is equal to the product of their determinants.
Addition property: The determinant of the sum of two matrices is equal to the sum of their determinants.
Scalar multiplication property: The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the dimension of the matrix, multiplied by the determinant of the original matrix.

Absolute Value of a Determinant

The absolute value of a determinant is calculated by taking the absolute value of the determinant. In other words, if the determinant is positive, its absolute value is the same as the determinant. If the determinant is negative, its absolute value is the opposite of the determinant.

Problem: Determinant of a 2x2 Matrix

Given the following two matrices:

\left|\begin{array}{ll}2x & 5 \\ 12 & x\end{array}\right| = \left|\begin{array}{rr}6 & -5 \\ 4 & 3\end{array}\right|

we need to find the value of $x$ .

Solution

To solve this problem, we can start by calculating the determinants of the two matrices.

For the first matrix, the determinant is:

\left|\begin{array}{ll}2x & 5 \\ 12 & x\end{array}\right| = (2x)(x) - (5)(12) = 2x^2 - 60

For the second matrix, the determinant is:

\left|\begin{array}{rr}6 & -5 \\ 4 & 3\end{array}\right| = (6)(3) - (-5)(4) = 18 + 20 = 38

Since the two determinants are equal, we can set up the following equation:

2x^2 - 60 = 38

Simplifying the equation, we get:

2x^2 = 98

Dividing both sides by 2, we get:

x^2 = 49

Taking the square root of both sides, we get:

x = \pm 7

Therefore, the value of $x$ is $\pm 7$ .

Conclusion

In this article, we explored the concept of determinants of a 2x2 matrix and used it to solve a problem involving absolute values. We calculated the determinants of two matrices and set up an equation to find the value of $x$ . The solution to the problem is $x = \pm 7$ . This problem demonstrates the importance of determinants in mathematics and their applications in various fields.

Final Answer

The final answer is $\boxed{\pm 7}$ .

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Q&A: Determinant of a 2x2 Matrix

Q1: What is a determinant of a 2x2 matrix?

A1: A determinant of a 2x2 matrix is a scalar value that can be calculated from the elements of the matrix. It is calculated using the formula:

\left|\begin{array}{ll}a & b \\ c & d\end{array}\right| = ad - bc

where $a$ , $b$ , $c$ , and $d$ are the elements of the matrix.

Q2: What are the properties of determinants?

A2: Determinants have several important properties that make them useful in various applications. Some of the key properties include:

Multiplication property: The determinant of a product of two matrices is equal to the product of their determinants.
Addition property: The determinant of the sum of two matrices is equal to the sum of their determinants.
Scalar multiplication property: The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the dimension of the matrix, multiplied by the determinant of the original matrix.

Q3: How do you calculate the absolute value of a determinant?

A3: The absolute value of a determinant is calculated by taking the absolute value of the determinant. In other words, if the determinant is positive, its absolute value is the same as the determinant. If the determinant is negative, its absolute value is the opposite of the determinant.

Q4: How do you solve a problem involving determinants and absolute values?

A4: To solve a problem involving determinants and absolute values, you can start by calculating the determinants of the matrices involved. Then, you can set up an equation using the determinants and solve for the unknown variable.

Q5: What is the final answer to the problem involving determinants and absolute values?

A5: The final answer to the problem involving determinants and absolute values is $\boxed{\pm 7}$ .

Q6: What are some real-world applications of determinants?

A6: Determinants have numerous real-world applications in various fields, including physics, engineering, and computer science. Some examples include:

Linear algebra: Determinants are used to solve systems of linear equations and to find the inverse of a matrix.
Physics: Determinants are used to calculate the determinant of a matrix representing a physical system, such as a mechanical system or an electrical circuit.
Computer science: Determinants are used in computer graphics and game development to perform transformations and projections.

Q7: How do you use determinants to solve a system of linear equations?

A7: To use determinants to solve a system of linear equations, you can start by setting up a matrix representing the system of equations. Then, you can calculate the determinant of the matrix and use it to find the solution to the system.

Q8: What is the relationship between determinants and eigenvalues?

A8: The determinant of a matrix is related to its eigenvalues. Specifically, the determinant of a matrix is equal to the product of its eigenvalues.

Q9: How do you use determinants to find the inverse of a matrix?

A9: To use determinants to find the inverse of a matrix, you can start by calculating the determinant of the matrix. Then, you can use the determinant to find the inverse of the matrix.

Q10: What are some common mistakes to avoid when working with determinants?

A10: Some common mistakes to avoid when working with determinants include:

Not checking the sign of the determinant: Make sure to check the sign of the determinant before using it in a calculation.
Not using the correct formula: Make sure to use the correct formula for calculating the determinant of a matrix.
Not simplifying the determinant: Make sure to simplify the determinant before using it in a calculation.

Conclusion

In this article, we explored the concept of determinants of a 2x2 matrix and answered some common questions about determinants. We also discussed some real-world applications of determinants and provided some tips for avoiding common mistakes when working with determinants.