Question

If $P$ is a non null matrix of order $3$ with real number as entries, such that $P^3=0,$ where $O$ is null matrix of order $3$ then (where $I$ is the identity matrix of order $3$)

• A. det. of $(4P^2-2P+I)$ is a non zero number.
• B. $I-2P$ is an invertible matrix
• C. If matrix $P$ has all integer entries, then $|I-4P^2|=0$
• D. If matrix $P$ has all integer entries, the absolute value of $|I-4P^2|=1$

Answer 1

Answer: (A), (B), (D)

The correct options are
A det. of $(4P^2-2P+I)$ is a non zero number.
B $I-2P$ is an invertible matrix
D If matrix $P$ has all integer entries, the absolute value of $|I-4P^2|=1$

$(2P+I)(4P^2-2P+I)=8P^3+I^3=I$
Similarly $(I-2P)(I+2P+4P^2)=I$
if $P$ has integer entries, then $|I-2P|$ and $|I+2P|$ will be equal to either $1$ or $-1$.