https://file.aiquickdraw.com/mathbot/file/1733698786305ztq94d77.jpg

This problem requires finding the value of \( x \) such that the determinant of the matrix \( A \) is equal to \(-51\). Let's calculate the determinant and solve for \( x \).

The matrix \( A \) is:
\[
A = \begin{bmatrix}
x-2 & 4 & 2 \\
-2 & 1 & 3 \\
x-4 & 3 & -2
\end{bmatrix}
\]

The determinant of \( A \) can be expanded as:
\[
\det(A) = (x-2) \det\begin{bmatrix} 1 & 3 \\ 3 & -2 \end{bmatrix}
- 4 \det\begin{bmatrix} -2 & 3 \\ x-4 & -2 \end{bmatrix}
+ 2 \det\begin{bmatrix} -2 & 1 \\ x-4 & 3 \end{bmatrix}
\]

### Step 1: Compute the sub-determinants
1. \(\det\begin{bmatrix} 1 & 3 \\ 3 & -2 \end{bmatrix} = (1)(-2) - (3)(3) = -2 - 9 = -11\)

2. \(\det\begin{bmatrix} -2 & 3 \\ x-4 & -2 \end{bmatrix} = (-2)(-2) - (3)(x-4) = 4 - 3(x-4) = 4 - 3x + 12 = 16 - 3x\)

3. \(\det\begin{bmatrix} -2 & 1 \\ x-4 & 3 \end{bmatrix} = (-2)(3) - (1)(x-4) = -6 - (x-4) = -6 - x + 4 = -2 - x\)

### Step 2: Expand the determinant
\[
\det(A) = (x-2)(-11) - 4(16 - 3x) + 2(-2 - x)
\]
Simplify each term:
1. \((x-2)(-11) = -11x + 22\)
2. \(-4(16 - 3x) = -64 + 12x\)
3. \(2(-2 - x) = -4 - 2x\)

Combine these terms:
\[
\det(A) = -11x + 22 - 64 + 12x - 4 - 2x
\]
\[
\det(A) = (-11x + 12x - 2x) + (22 - 64 - 4)
\]
\[
\det(A) = -x - 46
\]

### Step 3: Solve for \( x \) when \( \det(A) = -51 \)
\[
-x - 46 = -51
\]
\[
-x = -51 + 46
\]
\[
-x = -5 \implies x = 5
\]

### Final Answer:
The value of \( x \) is **5** (Option d).

---

Would you like a detailed walkthrough of any step? 

Here are five related questions to expand your understanding:
1. How can we use cofactor expansion to calculate determinants for larger matrices?
2. Why does setting \(\det(A) = -51\) help us solve for \( x \)?
3. What are the properties of determinants that simplify matrix calculations?
4. How does a zero determinant relate to the invertibility of a matrix?
5. What happens if multiple values of \( x \) satisfy \(\det(A) = k\)?

**Tip:** Always double-check signs when calculating determinants, as a small error can lead to incorrect results!

¿Cuál es el valor de x tal que det(A) = -51?, donde A = [ [x-2, 4, 2], [-2, 1, 3], [x-4, 3, -2] ]

Learn how to find the value of x such that the determinant of a given matrix is -51. This detailed solution covers key concepts in linear algebra and determinants, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation