Proof on matrices :(

May 2015
30
0
Manila
I need help on proving the proof questions..
1. Let A be an n x n matrix in reduced row echelon form. If A is not equal to In (identity matrix), prove that A has a row consisting entirely of zeros.

2. If A = [a b which is a 2 x 2 matrix;
c d]
show that A is nonsingular if and only if ad-bc is not equal to zero
- So far I know that if the determinant is not equal to zero, then the inverse of A exists. I am stuck on the next step..

3. Find all values of a for which the inverse of the matrix exists. What is A inverse?
A= (1 1 0 ; 3 x 3 matrix
1 0 0
1 2 a)
Thanks in advance
 

Plato

MHF Helper
Aug 2006
22,507
8,664
I need help on proving the proof questions..
1. Let A be an n x n matrix in reduced row echelon form. If A is not equal to In (identity matrix), prove that A has a row consisting entirely of zeros.
2. If A = [a b which is a 2 x 2 matrix;
c d]
show that A is nonsingular if and only if ad-bc is not equal to zero
- So far I know that if the determinant is not equal to zero, then the inverse of A exists. I am stuck on the next step..

3. Find all values of a for which the inverse of the matrix exists. What is A inverse?
A= (1 1 0 ; 3 x 3 matrix
1 0 0
1 2 a)
Did I miss your showing your own work?
 
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May 2015
30
0
Manila
Sorry for late reply.
So far, this is what I've did.
1. We know that if A has a row consisting entirely of zeros then the determinant of A is zero which means an inverse does not exist.
I don't know what is the next step here.... (I need help here please...)
2. Recall that A inverse is A^-1=1/(ad-bc) [a b
c d] ==> Suppose if ad-bc was equal to zero the inverse would not exists because it would be undefined since 1/(ad-bc=0 does not exist). However, if the determinant was nonzero then it indicates that the inverse exists. Hence, if the determinant is not equal to zero, then the inverse of A exists.
Is this a valid statment for number 2?