# proof question regarding matrices

#### massmatics

I have a new question here.
3. Find all values of a for which the inverse of the 3 x 3 matrix exists. What is A inverse?
A = (1 1 0
1 0 0
1 2 a)

Here is my work
So far I know that if det A is nonzero, then A inverse exists. So I used
A = (1 1 0|1 1
1 0 0|1 0
1 2 a|1 2)
and then I used the shortcut method diagonally. |A| = 1(0)(a)+1(0)(1)+0(1)(2)-0(0)(1)-1(0)(2)-1(1)(a) != 0
So I got -a != 0 so I concluded that a is all real numbers except 0... Is that correct?

#### Idea

$$\displaystyle a\neq 0$$ is correct

next, find $$\displaystyle A^{-1}$$

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#### massmatics

Thanks a lot Can I also ask for help in another proof question?
1. Let A be an n x n matrix in reduced row echelon form (rref). If A is not equal to In (identity matrix), prove that A has a row consisting entirely of zeros.
My solution:
Definition of reduced row echelon form states that all nonzero rows (rows with at least 1 nonzero element) are above any rows of all zeros (all zero rows, if any, belong at the bottom of the matrix).
Since A is unequal to matrix identity, we can also rewrite A in rref A=(1 0
0 0)
Since 1st row is a nonzero row the 2nd row should be a row of zeros when A unequal to identity matrix. Thus, A has a row of consisting entirely of zeros (Is this a valid proof?...)