If the determinant
`|(x, 1, 3),(0, 0,1),(1, x, 4)|=0`
then what is x equal to ?

Options

A. `-2 or 2 `
B. `-3 or 3 `
C. `-1 or 1 `
D. 3 or 4

Answer

To solve the problem, we need to evaluate the determinant of the given matrix and set it equal to zero. The matrix is:

\[ \begin{vmatrix} x & 1 & 3 \\ 0 & 0 & 1 \\ 1 & x & 4 \end{vmatrix} \]

### Step 1: Calculate the determinant

Using the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

- \( a = x, b = 1, c = 3 \)
- \( d = 0, e = 0, f = 1 \)
- \( g = 1, h = x, i = 4 \)

Substituting these values into the determinant formula:

\[ \text{det}(A) = x(0 \cdot 4 - 1 \cdot x) - 1(0 \cdot 4 - 1 \cdot 1) + 3(0 \cdot x - 0 \cdot 1) \]

Calculating each term:

1. The first term: \( x(0 - x) = -x^2 \)
2. The second term: \( -1(0 - 1) = 1 \)
3. The third term: \( 3(0 - 0) = 0 \)

Putting it all together:

\[ \text{det}(A) = -x^2 + 1 + 0 = -x^2 + 1 \]

### Step 2: Set the determinant equal to zero

Now we set the determinant equal to zero:

\[ -x^2 + 1 = 0 \]

### Step 3: Solve for \( x \)

Rearranging the equation gives:

\[ -x^2 = -1 \]

Multiplying both sides by -1:

\[ x^2 = 1 \]

Taking the square root of both sides:

\[ x = \pm 1 \]

### Conclusion

Thus, the values of \( x \) are:

\[ x = 1 \quad \text{or} \quad x = -1 \]

### Final Answer

The values of \( x \) are \( x = 1 \) and \( x = -1 \).

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