Consider the following augmented matrix(0 2aFiud the conditions On a such that the system has no slution...

Let $A$ be a $2 \times 2$ matrix given by $$ A=\left[\begin{array}{ll} x & 0 \\ 0 & y \end{array}\right] $$ Use the determinant of $A$ to determine the conditions under which $A^{-1}$ exists.

So they've given us three different Matrix sees in this problem, and they want us to described thes solutions to the Matrix. So in this very 1st 1 here we have this bottom row here that is 000 That means zero variables equals zero. This is a true statement. So because of that, we can look at the 1st 1 So that's saying an X and two y's egos equal to negative four. So because we don't know what X and we don't know why it is, we would say that this solution system is dependent. It be dependent on whatever exes or whatever. Why is to tell us what the other value would have to be in the 2nd 1? Here, we've got 106 that tells us that X is equal to six, and then we have 010 So it's telling us that why is equal to zero. That means this one has a single solution. It's not dependent X will always be six. Why will always be zero for this one toe workout? When we look at the 3rd 1 we end up with zero variables is equal to two. Now this statement we consider false, because whenever we set up these kind of equations, we always put the variables on the one side to make the equation and all of the Constance get moved over to our answer side. So if we have no variables, we should have no answer. So because we have an answer, this is a false statement, which means for this one, we can't have a solution. So any time there's a false statement included in the Matrix, that must mean there is no solution.

This is a little question Number 63 as we can see here when a 0.0, then the last ruler of zero. And there would be a no pilot entry at the third column, right? When is equal to zero. And the system has infinitely many solutions then financially Manisha, right? No. When is not equal to zero then against that? That the system is consistent and stuff has only one solution right on the one solution that is negative. 25 and zero. Right. Thank you.