Solving For X In Determinant Equations A Step By Step Guide

In the realm of mathematics, solving for unknown variables is a fundamental skill. This article delves into a specific type of problem: solving for x within a determinant equation. Determinants, often represented by vertical bars enclosing a matrix, are scalar values derived from square matrices. Understanding how to calculate determinants and manipulate equations involving them is crucial in various fields, including linear algebra, calculus, and physics. This comprehensive guide will walk you through the process of solving the given determinant equation, providing a clear, step-by-step explanation suitable for learners of all levels. We'll break down the equation, explore the properties of determinants, and apply the necessary techniques to isolate x and find its value. So, let's embark on this mathematical journey and unravel the solution together!

The problem we aim to solve is:

$$\left|\begin{array}{ccc} -5 & 1 & -2 \\ 3 & 2 x & -5 \\ 4 & 0 & 5 \end{array}\right|=67$$

Our goal is to determine the value of x that satisfies this equation. This requires a solid understanding of how determinants are calculated and how algebraic manipulations can be used to isolate the variable.

Before we dive into solving the equation, let's solidify our understanding of determinants. A determinant is a scalar value that can be computed from the elements of a square matrix. For a 3x3 matrix, the determinant can be calculated using various methods, including the expansion by minors (also known as cofactor expansion). This method involves selecting a row or column, and then expanding along that row or column using the minors and cofactors of the elements. Let's consider a general 3x3 matrix:

$$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$

The determinant of this matrix, denoted as |A|, can be calculated as follows:

|A| = a(ei - fh) - b(di - fg) + c(dh - eg)

This formula may seem daunting at first, but it's simply a systematic way of multiplying and adding elements of the matrix. Each term in the expansion corresponds to an element in the first row (a, b, c) multiplied by its cofactor. The cofactor is the determinant of the 2x2 matrix obtained by deleting the row and column containing that element, multiplied by a sign (+ or -) that alternates. Understanding this formula is key to solving determinant equations.

Key Properties of Determinants

Several properties of determinants are essential for simplifying calculations and solving equations. These include:

• Determinant of a Transpose: The determinant of a matrix is equal to the determinant of its transpose.
• Row or Column Swap: Swapping two rows or two columns of a matrix changes the sign of the determinant.
• Scalar Multiplication: Multiplying a row or column by a scalar multiplies the determinant by that scalar.
• Zero Row or Column: If a matrix has a row or column of zeros, its determinant is zero.
• Identical Rows or Columns: If a matrix has two identical rows or columns, its determinant is zero.

These properties can be used strategically to simplify determinant calculations and solve equations more efficiently. For example, if we can manipulate a matrix to have a row or column of zeros, we immediately know that the determinant is zero. Similarly, if we can identify two identical rows or columns, we can conclude that the determinant is zero without further calculation. Mastering these properties is crucial for tackling more complex determinant problems.

Now, let's apply our knowledge of determinants to the given equation. We need to calculate the determinant of the 3x3 matrix:

$$\begin{bmatrix} -5 & 1 & -2 \\ 3 & 2 x & -5 \\ 4 & 0 & 5 \end{bmatrix}$$

We'll use the expansion by minors method, expanding along the first row. This gives us:

Determinant = -5 * Determinant of $\begin{bmatrix} 2x & -5 \\ 0 & 5 \end{bmatrix}$ - 1 * Determinant of $\begin{bmatrix} 3 & -5 \\ 4 & 5 \end{bmatrix}$ + (-2) * Determinant of $\begin{bmatrix} 3 & 2x \\ 4 & 0 \end{bmatrix}$

Now, let's calculate the determinants of the 2x2 matrices:

• Determinant of $\begin{bmatrix} 2x & -5 \\ 0 & 5 \end{bmatrix}$ = (2x * 5) - (-5 * 0) = 10x
• Determinant of $\begin{bmatrix} 3 & -5 \\ 4 & 5 \end{bmatrix}$ = (3 * 5) - (-5 * 4) = 15 + 20 = 35
• Determinant of $\begin{bmatrix} 3 & 2x \\ 4 & 0 \end{bmatrix}$ = (3 * 0) - (2x * 4) = -8x

Substituting these values back into the expression for the determinant, we get:

Determinant = -5 * (10x) - 1 * (35) + (-2) * (-8x) = -50x - 35 + 16x

Simplifying this expression, we have:

Determinant = -34x - 35

This result is a linear expression in terms of x. It represents the value of the determinant for any given value of x. Now, we can use this expression to solve the original equation.

We now have an expression for the determinant in terms of x: -34x - 35. The original equation states that this determinant must equal 67. Therefore, we can set up the following equation:

-34x - 35 = 67

This is a simple linear equation that we can solve for x. To isolate x, we first add 35 to both sides of the equation:

-34x = 67 + 35

-34x = 102

Next, we divide both sides by -34:

x = 102 / -34

x = -3

Thus, the solution to the equation is x = -3. This means that when we substitute -3 for x in the original matrix and calculate the determinant, we will obtain the value 67. To verify our solution, we can substitute x = -3 back into the determinant expression and check if it equals 67.

To ensure the accuracy of our solution, let's substitute x = -3 back into the expression for the determinant and verify that it equals 67.

Determinant = -34x - 35

Substituting x = -3, we get:

Determinant = -34 * (-3) - 35

Determinant = 102 - 35

Determinant = 67

As we can see, the determinant is indeed equal to 67 when x = -3. This confirms that our solution is correct. Verification is an essential step in problem-solving, as it helps to catch any potential errors and ensures that the final answer is accurate. In this case, our verification process provides confidence in our solution and demonstrates a thorough understanding of the problem.

We can also substitute x = -3 back into the original matrix and calculate the determinant directly to further verify our result:

$$\left|\begin{array}{ccc} -5 & 1 & -2 \\ 3 & 2 * (-3) & -5 \\ 4 & 0 & 5 \end{array}\right| = \left|\begin{array}{ccc} -5 & 1 & -2 \\ 3 & -6 & -5 \\ 4 & 0 & 5 \end{array}\right|$$

Expanding along the first row, we get:

Determinant = -5 * ((-6 * 5) - (-5 * 0)) - 1 * ((3 * 5) - (-5 * 4)) + (-2) * ((3 * 0) - (-6 * 4))

Determinant = -5 * (-30) - 1 * (15 + 20) + (-2) * (24)

Determinant = 150 - 35 - 48

Determinant = 67

This further confirms that our solution x = -3 is correct.

In this article, we successfully solved for x in a determinant equation. We began by understanding the concept of determinants and their properties. We then calculated the determinant of the given 3x3 matrix, obtaining a linear expression in terms of x. By setting this expression equal to the given value (67), we formed a linear equation and solved for x. Finally, we verified our solution by substituting it back into the original equation and confirming that the determinant equals 67. This step-by-step approach demonstrates a clear and methodical way to solve determinant equations.

This problem highlights the importance of understanding determinants and their properties in linear algebra. The ability to calculate determinants and solve equations involving them is a valuable skill in various mathematical and scientific contexts. By mastering these concepts, you'll be well-equipped to tackle more complex problems in linear algebra and related fields. Remember, practice is key to success in mathematics. The more you practice solving determinant equations, the more comfortable and confident you'll become in your abilities.

Key Takeaways

• Determinants are scalar values calculated from square matrices.
• The expansion by minors method is a common technique for calculating determinants.
• Understanding the properties of determinants can simplify calculations.
• Solving determinant equations involves algebraic manipulation and equation solving skills.
• Verification is crucial to ensure the accuracy of the solution.

By following this guide and practicing regularly, you can confidently solve for variables within determinant equations and further enhance your mathematical prowess.