Matrices HELP!!!!!

[MathsIsFun]

Hi all,

I got a question in my textbook and i dont quite understand it:
It says:

For real numbers, The Null Factor law says that xy = 0 (x,y E R), then x=0 or y=0 or both
for matrices, however this law does not apply; there are infinitely many matrices for which AB = 0
Given Matrices A= [ a b ] and B = [ e f ]
...........................[ c d ] [ g h ]

use the properties of determinants to prove that if AB=0
then ad=bc

Thanks in advance

 

[Ishuda]

Hi all,

I got a question in my textbook and i dont quite understand it:
It says:

For real numbers, The Null Factor law says that xy = 0 (x,y E R), then x=0 or y=0 or both
for matrices, however this law does not apply; there are infinitely many matrices for which AB = 0
Given Matrices A= [ a b ] and B = [ e f ] use the properties of determinants to prove that AB=0
[ c d ] [ g h ]

Thanks in advance

I'm sorry, I don't quite understand. Do you mean that given A and B are two 2X2 matrices, if AB=0 then either det(A)=0 or det(B)=0 or both?

Prove this for
\(\displaystyle A = \begin{pmatrix}a& b\\c& d\end{pmatrix}\)
and
\(\displaystyle B = \begin{pmatrix}e& f\\g& h\end{pmatrix}\)
using the properties of determinants.

 

[MathsIsFun]

I'm sorry, I don't quite understand. Do you mean that given A and B are two 2X2 matrices, if AB=0 then either det(A)=0 or det(B)=0 or both?

Prove this for
\(\displaystyle A = \begin{pmatrix}a& b\\c& d\end{pmatrix}\)
and
\(\displaystyle B = \begin{pmatrix}e& f\\g& h\end{pmatrix}\)
using the properties of determinants.

Yes, But i dont know how to do this

 

[Ishuda]

Yes, But i dont know how to do this

Hint:
det(AB)=0
det(AB) = det(A) det(B) = (?) * (?) = 0

 

[MathsIsFun]

Hint:
det(AB)=0
det(AB) = det(A) det(B) = (?) * (?) = 0

I still dont get how to prove by using the properties of determinants