Question

If \(P\) and \(A\) are nonsingular matrices such that \[ |P|=|A|=4, \qquad |A+4I|=14, \] then \[ \frac{\left|P^{-1}AP+|A|I\right|} {|A|+|P|^{-1}} = \]

• A. \(\dfrac72\)
• B. \(2\)
• C. \(14\)
• D. \(\dfrac{29}{7}\)

Hint:

For any invertible matrix \(P\), \[ \boxed{ |P^{-1}MP| = |M| } \] because \[ |P^{-1}||P|=1. \] Thus, similar matrices always have the same determinant.

Answer 1

The Correct Option is B