Question

If $|\begin{matrix}1& 0& 0\\ x& x+2& 0\\ x^{2}& x& x+3\end{matrix}|=0$, then values of $x$ are ________.

• A. 2, 3
• B. -2, 3
• C. -2, -3
• D. 1, 2, 3
• E. -1, 2, -3

Hint:

Triangular matrix determinant = product of diagonal entries.

Answer 1

The Correct Option is C

To solve the determinant of the given matrix and find the values of \( x \) for which this determinant equals zero, let's first express the matrix and calculate its determinant:

The matrix is:

\[\begin{bmatrix} 1 & 0 & 0\\ x & x+2 & 0\\ x^{2} & x & x+3 \end{bmatrix}\]

To find the determinant of this matrix \(|A|\), we can use the expansion along the first row:

\[|A| = 1 \cdot ((x+2)(x+3) - 0 \cdot x) - 0 \cdot |C_{12}| + 0 \cdot |C_{13}|\]

So, the determinant simplifies to:

\[|A| = (x+2)(x+3)\]

We are given that:

\[(x+2)(x+3) = 0\]

Setting each factor to zero gives us the potential values of \( x \):

1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( x + 3 = 0 \) → \( x = -3 \)

Therefore, the values of \( x \) for which the determinant is zero are \( -2 \) and \( -3 \).

Thus, the correct answer is:

-2, -3