Advanced book on Mathematics Olympiad

68 2 Algebra

Z=

(

A 0

CIn

)

.

According to the rule of Laplace, the determinant ofZcan be computed by expanding
along then×nminors from the topnrows, and all of them are zero except for the
first. Hence detZ =detA·detIn =detA, and so the matricesX, Y, Zsolve the
problem. 

214.Show that if

x=

∣∣

∣∣ab

cd

∣∣

∣∣ and x′=

∣∣

∣∣a

′b′

c′d′

∣∣

∣∣,

then

(xx′)^2 =

∣∣

∣

∣∣

∣∣

∣

ab′cb′ba′da′

ad′cd′bc′dc′

bb′db′aa′ca′

bd′dd′ac′cc′

∣∣

∣

∣∣

∣∣

∣

.

215.LetA, B, C, Dben×nmatrices such thatAC=CA. Prove that

det

(

AB

CD

)

=det(AD−CB).

216.LetXandYben×nmatrices. Prove that

det(In−XY )=det(In−YX).

A property exploited often in Romanian mathematics competitions states that for any

n×nmatrixAwith real entries,

det(In+A^2 )≥ 0.

The proof is straightforward:

det(In+A^2 )=det((In+iA)(In−iA))=det(In+iA)det(In−iA)

=det(In+iA)det(In+iA)=det(In+iA)det(In+iA).

In this computation the bar denotes the complex conjugate, and the last equality follows

from the fact that the determinant is a polynomial in the entries. The final expression is

positive, being equal to|det(In+iA)|^2.

Use this property to solve the following problems, while assuming that all matrices

have real entries.

217.LetAandBben×nmatrices that commute. Prove that if det(A+B)≥0, then

det(Ak+Bk)≥0 for allk≥1.