If the trace of the matrix `A= [{:( x-1 ,0,2,5),( 3, x^(2) - 2 ,4,1),( -1,-2,x-3,1),(2,0,4,x^(2)-6) :}]` is 0 then x is equal to

Options

A. ` (-2,3) `
B. ` (2,-3) `
C. ` ( -3,2) `
D. `( 3,-2) `

Answer

To find the value of \( x \) such that the trace of the matrix \( A \) is 0, we can follow these steps:

### Step 1: Define the Trace of the Matrix
The trace of a matrix is the sum of its diagonal elements. For the given matrix \( A \):
\[ A = \begin{pmatrix} x - 1 & 0 & 2 & 5 \\ 3 & x^2 - 2 & 4 & 1 \\ -1 & -2 & x - 3 & 1 \\ 2 & 0 & 4 & x^2 - 6 \end{pmatrix} \]
The diagonal elements are:
1. \( x - 1 \)
2. \( x^2 - 2 \)
3. \( x - 3 \)
4. \( x^2 - 6 \)

### Step 2: Write the Equation for the Trace
We need to set the sum of these diagonal elements equal to 0:
\[ (x - 1) + (x^2 - 2) + (x - 3) + (x^2 - 6) = 0 \]

### Step 3: Simplify the Equation
Combine like terms:
\[ x - 1 + x^2 - 2 + x - 3 + x^2 - 6 = 0 \]
This simplifies to:
\[ 2x^2 + 2x - 12 = 0 \]

### Step 4: Divide the Equation by 2
To simplify further, divide the entire equation by 2:
\[ x^2 + x - 6 = 0 \]

### Step 5: Factor the Quadratic Equation
Now, we can factor the quadratic equation:
\[ x^2 + 3x - 2x - 6 = 0 \]
This can be factored as:
\[ (x + 3)(x - 2) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 3 = 0 \) → \( x = -3 \)
2. \( x - 2 = 0 \) → \( x = 2 \)

### Conclusion
The values of \( x \) that make the trace of the matrix \( A \) equal to 0 are:
\[ x = -3 \quad \text{and} \quad x = 2 \]