Find X: Determinant Of Matrix A = 15
Hey guys! Let's dive into a cool math problem today where we need to figure out the value of 'x' in a matrix so that its determinant equals 15. Buckle up, because we're about to get our math hats on and solve this thing together!
Understanding Determinants
Before we jump into solving for 'x', let's quickly recap what a determinant is. In the simplest terms, the determinant is a special number that can be calculated from a square matrix (a matrix with the same number of rows and columns). It provides crucial information about the matrix, such as whether the matrix has an inverse (which is super useful in solving systems of equations) and the volume scaling factor when the matrix represents a linear transformation.
For a 2x2 matrix like the one we have, calculating the determinant is straightforward. If we have a matrix:
| a b | | c d |
The determinant (often written as det(A) or |A|) is calculated as:
det(A) = ad - bc
So, we multiply the elements on the main diagonal (top-left to bottom-right) and subtract the product of the elements on the other diagonal (top-right to bottom-left). Simple, right?
Why Determinants Matter
You might be wondering, “Okay, we can calculate this number, but why bother?” Well, the determinant is a powerful tool in various areas of mathematics and its applications. Here are a few reasons why determinants are so important:
- Invertibility: A matrix has an inverse if and only if its determinant is not zero. This is fundamental in solving systems of linear equations. If the determinant is zero, the matrix is singular, and the system either has no solution or infinitely many solutions.
- Linear Transformations: In the context of linear transformations, the absolute value of the determinant represents the factor by which the transformation scales areas (in 2D) or volumes (in 3D). A determinant of 2, for example, means that areas are doubled under the transformation.
- Eigenvalues and Eigenvectors: Determinants play a crucial role in finding eigenvalues, which are characteristic values associated with a matrix. Eigenvalues and eigenvectors are essential in many applications, including stability analysis in physics and engineering.
- Geometric Interpretation: The determinant can also be interpreted geometrically. In 2D, the determinant of a 2x2 matrix formed by two vectors represents the signed area of the parallelogram spanned by those vectors. In 3D, the determinant of a 3x3 matrix represents the signed volume of the parallelepiped spanned by three vectors.
Now that we appreciate the importance of determinants, let's get back to our problem and find the value of 'x'.
Setting Up the Equation
We're given the matrix A:
| x 2 | | 3 4 |
And we know that the determinant of A, det(A), is equal to 15. Using the formula we just discussed, we can write the equation for the determinant:
det(A) = (x * 4) - (2 * 3) = 15
Let's simplify this:
4x - 6 = 15
Now we have a simple linear equation to solve for 'x'.
Solving for x
To isolate 'x', we'll first add 6 to both sides of the equation:
4x - 6 + 6 = 15 + 6
4x = 21
Next, we'll divide both sides by 4:
4x / 4 = 21 / 4
x = 21 / 4
So, the value of x that makes the determinant of matrix A equal to 15 is 21/4. But wait! That's not one of the options provided. Let's double-check our work to make sure we didn't make a mistake.
Double-Checking Our Work
It's always a good idea to double-check your work, especially in math problems. Let's go back to our equation:
4x - 6 = 15
We added 6 to both sides, which gave us:
4x = 21
Then we divided by 4, resulting in:
x = 21 / 4
Okay, our calculations seem correct. It looks like there might be a mistake in the answer choices provided in the original question. The correct value of x is indeed 21/4. None of the options (A) 1/4, (B) 0, (C) 8, (D) -1/4, or (E) 2 are correct.
Analyzing the Answer Choices
Just for fun, let's see what the determinant would be if we plugged in the given answer choices:
- A) x = 1/4: det(A) = (1/4 * 4) - (2 * 3) = 1 - 6 = -5
- B) x = 0: det(A) = (0 * 4) - (2 * 3) = 0 - 6 = -6
- C) x = 8: det(A) = (8 * 4) - (2 * 3) = 32 - 6 = 26
- D) x = -1/4: det(A) = (-1/4 * 4) - (2 * 3) = -1 - 6 = -7
- E) x = 2: det(A) = (2 * 4) - (2 * 3) = 8 - 6 = 2
As we can see, none of these values give us a determinant of 15. This further confirms that the correct answer, x = 21/4, is not among the options.
Conclusion
So, there you have it! We've successfully found the value of 'x' that makes the determinant of matrix A equal to 15. The correct value is x = 21/4, which wasn't one of the provided options. This is a great reminder that sometimes, the answer choices can be misleading, and it's crucial to trust your calculations and understanding of the concepts.
Remember, determinants are powerful tools in linear algebra, and mastering them can open doors to solving a wide range of problems. Keep practicing, and you'll become a determinant whiz in no time! Keep your mathematical abilities sharp and never hesitate to double-check your work.
I hope you guys found this explanation helpful and engaging. Math can be fun, especially when we break it down step by step. Keep up the great work, and I'll catch you in the next problem-solving adventure! Stay curious and keep exploring! This exploration of determinant properties is a fundamental concept in linear algebra.
