I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:

If two singular matrices $A, B$ exist s.t. the determinant of $EA-hB$ is identically zero for all complex scalars $h$ and all real matrices $E$ then either $A=YB$ or $B=ZA$, $Y$ and $Z$ being undetermined matrices.

Is it true (vacuously or not) in general?

I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:

If two singular matrices $A, B$ exist s.t. the determinant of $EA-B$ is identically zero for all real matrices $E$, then either $A=YB$ or $B=ZA$, $Y$ and $Z$ being undetermined matrices.

Is it true (vacuously or not) in general?

I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:

If two singular matrices $A, B$ exist s.t. the determinant of $EA-hB$ is identically zero for all complex scalars $h$ and all real matrices $E$ then either matrix must be a matrix multiple, left or right, of the other.

Is it true (vacuously or not) in general?

I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:

If two singular matrices $A, B$ exist s.t. the determinant of $EA-hB$ is identically zero for all complex scalars $h$ and all real matrices $E$ then either $A=YB$ or $B=ZA$, $Y$ and $Z$ being undetermined matrices.

Is it true (vacuously or not) in general?