If the trace of the matrix `A=({:(x-5,0,2,4),(3,x^(2)-10,6,1),(-2,3,x-7,1),(1,2,0,-2):})` assumes the value zero, then the value of x equals to

Options

A. `-6, -4`
B. `-6, 4`
C. 6, 4
D. `6, -4`

Answer

To find the value of \( x \) such that the trace of the matrix \( A \) is zero, we will follow these steps:

1. **Identify the matrix**: The given matrix is
\[ A = \begin{pmatrix} x - 5 & 0 & 2 & 4 \\ 3 & x^2 - 10 & 6 & 1 \\ -2 & 3 & x - 7 & 1 \\ 1 & 2 & 0 & -2 \end{pmatrix} \]

2. **Calculate the trace**: The trace of a matrix is the sum of its diagonal elements. For matrix \( A \), the diagonal elements are:
- First element: \( x - 5 \)
- Second element: \( x^2 - 10 \)
- Third element: \( x - 7 \)
- Fourth element: \( -2 \)

Therefore, the trace \( \text{tr}(A) \) is given by:
\[ \text{tr}(A) = (x - 5) + (x^2 - 10) + (x - 7) + (-2) \]

3. **Combine like terms**: Simplifying the expression for the trace:
\[ \text{tr}(A) = x - 5 + x^2 - 10 + x - 7 - 2 \]
\[ = x^2 + (x + x) + (-5 - 10 - 7 - 2) \]
\[ = x^2 + 2x - 24 \]

4. **Set the trace to zero**: We want the trace to equal zero:
\[ x^2 + 2x - 24 = 0 \]

5. **Factor the quadratic equation**: We need to factor \( x^2 + 2x - 24 \). We look for two numbers that multiply to \(-24\) and add up to \(2\). The numbers \(6\) and \(-4\) work:
\[ (x + 6)(x - 4) = 0 \]

6. **Solve for \( x \)**: Setting each factor to zero gives us:
\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

Thus, the possible values of \( x \) are \( -6 \) and \( 4 \).

### Final Answer:
The values of \( x \) are \( -6 \) and \( 4 \).