Question

Let
\(M=\begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 2 & 0 & 0 & 0 & -4 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 3 \\ 0 & 0 & 0 & 2 & 2 \end{pmatrix}\)
If p(x) is the characteristic polynomial of M, then p(2) - 1 equals _________

Answer 1

Correct Answer: 31

To find \( p(2) - 1 \), we first need to calculate the characteristic polynomial of the matrix \( M \). The characteristic polynomial is given by the determinant of \( M - xI \), where \( I \) is the identity matrix and \( x \) is the variable. For this matrix, we calculate the determinant of \( M - 2I \), substitute \( x = 2 \) into the resulting polynomial, and subtract 1 to obtain the value \( p(2) - 1 = 31 \). Thus, the correct answer is 31.