Advanced book on Mathematics Olympiad

2.3 Linear Algebra 73

2.3.4 Systems of Linear Equations...............................

A system ofmlinear equations withnunknowns can be written as

Ax=b,

whereAis anm×nmatrix called the coefficient matrix, andbis anm-dimensional

vector. Ifm=n, the system has a unique solution if and only if the coefficient matrixA

is invertible. IfAis not invertible, the system can have either infinitely many solutions

or none at all. If additionallyb=0, then the system does have infinitely many solutions

and the codimension of the space of solutions is equal to the rank ofA.

We illustrate this section with two problems that apparently have nothing to do with

the topic. The first was published inMathematics Gazette, Bucharest,byL.Pîr ̧san.

Example.Consider the matrices

A=

(

ab

cd

)

,B=

(

αβ

γδ

)

,C=

⎛

⎜

⎜

⎝

aα bβ aγ bγ

aβ bβ aδ bδ

cα dα cγ dγ

cβ dβ cδ d δ

⎞

⎟

⎟

⎠,

wherea, b, c, d, α, β, γ, δare real numbers. Prove that ifAandBare invertible, thenC

is invertible as well.

Solution.Let us consider the matrix equationAXB=D, where

X=

(

xz

yt

)

and D=

(

mn

pq

)

.

Solving it forXgivesX=A−^1 DB−^1 , and soXis uniquely determined byA,B, and

D. Multiplying out the matrices in this equation,

(

ab

cd

)(

xz

yt

)(

αβ

γδ

)

=

(

mn

pq

)

,

we obtain

(

aαx+bαy+aγ z+bγ t aβx+bβy+aδz+bδt

cαx+dαy+cγ z+dγt cβx+dβy+cδz+dδt

)

=

(

mn

pq

)

.

This is a system in the unknownsx, y, z, t:

aαx+bαy+aγ z+bγ t=m,

aβx+bβy+aδz+bδt=n,