Question

Let
A =\(\begin{pmatrix}     4 & -2 \\     \alpha & \beta \\ \end{pmatrix}\)
If A² + γA + 18I = 0, then det (A) is equal to ______.

• A. –18
• B. 18
• C. –50
• D. 50

Answer 1

The Correct Option is B

To solve this problem, we start by using the given equation for matrix \( A \):

\( A^2 + \gamma A + 18I = 0 \)

Given the matrix \( A = \begin{pmatrix} 4 & -2 \\ \alpha & \beta \\ \end{pmatrix} \), we need to find its determinant. The characteristic equation of a matrix \( A \) is of the form:

\( A^2 - (\text{trace of } A)A + (\text{det of } A)I = 0 \)

Matching this with the given equation \( A^2 + \gamma A + 18I = 0 \), we equate the coefficients to find the determinant. The trace of matrix \( A \) is given by:

\( \text{trace of } A = 4 + \beta \)

Thus, comparing the coefficients:

• \(\gamma = -(\text{trace of } A)\)
• \(18 = \text{det of } A\)

This directly leads us to the conclusion that:

\(\text{det of } A = 18\)

Therefore, the determinant of matrix \( A \) is 18.