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

Answer 1

The Correct Option is A

Given data:
Let, A be an invertible \(4×4\) matrix with complex entries. 

The determinant of A is denoted as \(\text{det}(A),\) and the adjoint of \(A\) is denoted as \(\text{adj}(A).\)

Given that \(\text{det}(\text{adj}(A)) = 5\), 

Using the property of determinants:

\(det(A × adj(A)) = det(A)^{(n-1)}\)

where \(n\) is the size of the square matrix (in this case, \(n = 4\)).

We have \(A × \text{adj}(A) = \text{det}(A) × I\), (where \(I\) is the identity matrix.)

So, det(A) × det(adj(A)) = d

Substituting det(adj(A)) = 5 and n = 4:

\(det(A) × 5 = det(A)^3\)

Now, solve for \(det(A)\):

\(det(A)^3 - 5×det(A) = 0\)

\(⇒det(A) × (det(A)^2 - 5) = 0\)

\(det(A) = 0\) or \(det(A)^2 - 5 = 0\)

If \(det(A) = 0\) : If \(det(A) = 0\), then A is a singular matrix, and it won't be invertible. However, we are given that A is an invertible matrix, so this case is not possible.

If \(det(A)^2 - 5 = 0: det(A)^2 = 5.\)

\(|det(A)| = √(5)\)

So, considering the positive values for \(det(A)\) are \(+√(5)\).

Therefore, the number of possible values of the determinant of \(A\) is \(1\).(_Ans)