Equivalent Conditions for Nonsingularity (Matrices)

[jtruth914]

True or False. If true explain or prove answer, and if false give an example to show the statement is not always true.

1. If A is a 4x4 matrix and a1+a2=a3+2a4, then A must be singular.
2. If A is row equivalent to both B and C, then A is row equivalent to B+C.

My Work:
1. I say it's False because A is nonsingular. But I don't know how to show an example of it.
2. I say it's False. I know that If A is row equivalent to B, and B is row equivalent to C, then A is row equivalent to C. I don't know how to show an example for 2 neither.

 

[Ray Vickson]

True or False. If true explain or prove answer, and if false give an example to show the statement is not always true.

1. If A is a 4x4 matrix and a1+a2=a3+2a4, then A must be singular.
2. If A is row equivalent to both B and C, then A is row equivalent to B+C.

My Work:
1. I say it's False because A is nonsingular. But I don't know how to show an example of it.
2. I say it's False. I know that If A is row equivalent to B, and B is row equivalent to C, then A is row equivalent to C. I don't know how to show an example for 2 neither.

What do you mean by a1, etc?

 

[jtruth914]

I think a1 is referring to the entries in matrix A.

 

[Ray Vickson]

I think a1 is referring to the entries in matrix A.

If that is so then the question makes no sense. The matrix has 16 entries, so which 4 of the 16 are a1, a2, a3 and a4?

 

[Office_Shredder]

I would guess that a1, a2, a3 and a4 are referring to either the rows or columns of the matrix, but you'll have to fill us in on what your notation is.

 

[jtruth914]

The book uses that notation to refer to the column.

 

[vela]

In your answer to #1, you simply assert A is non-singular. How do you know this?