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The question you are asking to solve is part (গ) from the image, which refers to finding the trace of matrix \( A \) under the condition \( \det(A) = 0 \). Let's break it down.

We are given the matrix \( A \) as:

\[
A = \begin{pmatrix}
3 + x & 4 & 1 \\
4 & 1 + x & 3 \\
1 & 3 & 4 + x
\end{pmatrix}
\]

The trace of a matrix is the sum of its diagonal elements. So, for matrix \( A \), the trace is:

\[
\text{Trace}(A) = (3 + x) + (1 + x) + (4 + x)
\]
\[
\text{Trace}(A) = 3 + 1 + 4 + x + x + x = 8 + 3x
\]

Now, we also have the condition \( \det(A) = 0 \). To solve for \( x \), we would need to calculate the determinant of matrix \( A \) and set it equal to zero. Let me proceed with calculating \( \det(A) \).

The determinant of a 3x3 matrix is calculated as:

\[
\det(A) = (3+x) \left( (1+x)(4+x) - (3 \cdot 3) \right) - 4 \left( 4(4+x) - (1 \cdot 3) \right) + 1 \left( 4 \cdot 3 - (1+x) \cdot 1 \right)
\]

Let me compute the determinant step by step.The determinant of matrix \( A \) is:

\[
\det(A) = x^3 + 8x^2 - 7x - 56
\]

We are given that \( \det(A) = 0 \), so we need to solve:

\[
x^3 + 8x^2 - 7x - 56 = 0
\]

Now, let's solve this cubic equation for \( x \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can solve the cubic equation \( x^3 + 8x^2 - 7x - 56 = 0 \) either by factoring or using a numerical method.

If you'd like further help or explanations on how to solve cubic equations, feel free to ask!

Here are some related questions you can explore:
1. How do you factor a cubic equation?
2. Can the determinant of matrix \( B \) be solved similarly to matrix \( A \)?
3. What is the significance of setting the determinant equal to zero?
4. How can trace help in solving eigenvalue problems?
5. How are determinants used in solving systems of linear equations?

**Tip:** For cubic equations, using numerical methods like Newton's method or synthetic division can speed up finding real roots.

Solve গ of question 3 from the provided image, involving finding the trace of matrix A under the condition det(A) = 0.

Learn how to find the trace of matrix A from the provided problem, where det(A) = 0. The solution explores key linear algebra concepts such as matrix determinants and traces, suitable for undergraduate students.

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