MathDB

matrices

Source: Romania District Olympiad 2013,grade XI(problem 2)

March 14, 2013

linear algebramatrixalgebra proposedalgebra

Problem Statement

Let the matrices of order 2 with the real elements

A

and

B

so that

AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}

and

\det \left( B \right)=2

. a) Prove that the matrix

A

is not invertible. b) Calculate

\det \left( A+2B \right)-\det \left( B+2A \right)