The system in matrix form is:
( 1 1 )( s )=( 25000 )
( 1 -1 )( d )=( 5000 )
Representing the equations:
s+d=25000
s-d=5000 where s=son’s share, d=daughter’s share.
Let matrix A=
( 1 1 )
( 1 -1 )
B=
( 25000 )
( 5000 )
and X=
( s )
( d )
So we have AX=B.
Therefore X=A⁻¹B.
A⁻¹=
½( 1 1 )
( 1 -1 )
So
( s ) = ½( 1 1 )( 25000 )
( d ) ( 1 -1 )( 5000 ) from which s=½(25000+5000)=15000
and d=½(25000-5000)=10000.
The son gets $15000 and the daughter $10000.
[To find the inverse of A we solve A×A⁻¹=I.
( 1 1 )( a b )=( 1 0 )
( 1 -1 )( c d ) ( 0 1 )
From this, a+c=1, a-c=0, b+d=0, b-d=1.
So a=c=½, d=-b, b=½, d=-½. So the inverse is:
½( 1 1 )
( 1 -1 ) or
-½( -1 -1)
( -1 1 ) where the scalar ½ or -½ is a common factor for the elements and can be taken outside the matrix.]