Question

Let A be an invertible matrix of size 4x4 with complex entries. If the determinant of adj(A) is 5,then the number of possible value of determinant of A is 

• A. \(3\)
• B. \(4\)
• C. \(6\)
• D. \(1\)
• E. \(2\)

Answer 1

The Correct Option is A

Given: \( A \) is a \( 4 \times 4 \) invertible matrix with complex entries, and \( \det(\text{adj}(A)) = 5 \).

We know the relationship between a matrix and its adjugate: \[ \text{adj}(A) = \det(A) \cdot A^{-1} \] Taking determinants of both sides: \[ \det(\text{adj}(A)) = \det(\det(A) \cdot A^{-1}) \] \[ = (\det(A))^4 \cdot \det(A^{-1}) \] \[ = (\det(A))^4 \cdot (\det(A))^{-1} \] \[ = (\det(A))^3 \]

Given \( \det(\text{adj}(A)) = 5 \), we have: \[ (\det(A))^3 = 5 \] The complex solutions are: \[ \det(A) = 5^{1/3}, \, 5^{1/3} \omega, \, 5^{1/3} \omega^2 \] where \( \omega = e^{2\pi i/3} \) is a primitive cube root of unity.

Thus there are 3 distinct possible values for \( \det(A) \).

Therefore, the number of possible values is 3.

Answer 2

Let \( A \) be a \( 4 \times 4 \) invertible matrix with complex entries. We are given that \( \det(\text{adj}(A)) = 5 \).

We know that for an invertible \( n \times n \) matrix \( A \), the following relation holds:

\[ A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \]

Taking the determinant of both sides, we get

\[ \det(A^{-1}) = \det\left( \frac{1}{\det(A)} \text{adj}(A) \right) \]

Since \( \det(A^{-1}) = \frac{1}{\det(A)} \), we have

\[ \frac{1}{\det(A)} = \frac{1}{(\det(A))^4} \det(\text{adj}(A)) \]

We are given that \( \det(\text{adj}(A)) = 5 \), and \( n=4 \). Thus,

\[ \frac{1}{\det(A)} = \frac{5}{(\det(A))^4} \] \[ (\det(A))^3 = 5 \]

Let \( x = \det(A) \). Then \( x^3 = 5 \). The solutions are the three cube roots of 5:

\[ x = \sqrt[3]{5}, \quad x = \sqrt[3]{5} \omega, \quad x = \sqrt[3]{5} \omega^2 \]

where \( \omega = e^{i\frac{2\pi}{3}} = \cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \) is a complex cube root of unity.

Therefore, there are 3 possible values for \( \det(A) \).