PROFESSOR: Hi, everyone.
So for the first part of this course, we've learned
basically the ins and outs of solving
linear systems of equations.
Today we're going to do a little review
of the basic concepts.
Hopefully we'll see a few of them in the following problem.
We're given a square matrix A-- a three by three square
matrix A-- where the last entry is a parameter k.
And this parameter will vary.
And we'll see what happens to the system of equations--
Ax equal to 2 3 7, for which k it has a unique solution, for
which k it has infinitely many solutions.
Then we'll find the LU decomposition.
And finally, we'll write down the complete
solution, the system.
So I'll give you a few minutes to work this
problem on your own.
And then please come back and see how I do it.
All right, welcome back.
So let's start with part A. For which k does this system,
Ax equal 2, 3, 7 have a unique solution?
So what do we know about square
systems of linear equations?
They have a unique solution when the matrix A is
invertible.
So now when is A invertible?
It is invertible when it is a full rank.
And how do we figure this out?
We do it by performing row operations.
We do it by doing eliminations on the matrix.
But since we want to simulate in exam setting, It always
pays off to see what tasks lie ahead of us.
So in part C, we're asked to find the LU decomposition.
This means that when we do row operations, we'd better keep
track what row operations we're doing exactly.
In particular, we'll write them down
as elementary matrices.
And in part D, we'll be asked to compute
the complete solution.
And therefore it's good to do row operations on the
augmented matrix A. So let's do this.
I'm going to write this.
The augmented matrix A is the following beast, 1, 1, 1, 1,
2, 3, 3, 4, k, and then 2, 3, 7.
So first thing, we subtract a multiple of row
one from row two.
And it's exactly negative 1 times the first
row plus the second.
Let me write down the corresponding elementary
matrix that does this.
It's E, 2, 1.
And it's lower diagonal with 1's on the diagonal.
And it's going to be exactly minus 1 in the first entry of
the second row.
So we get 1, 1, 1, 2, 0, 1, 2, 1.
And we copy down the third row.
Now we subtract a multiple of the first row
from the third one.
And let me write this here.
Yeah, we'll multiply the first row by negative 3, and add it
to the third one.
This is accomplished by the elementary matrix E, 3 1,
which is 1, 1, 1, negative three, and then 0, 0.
OK, 1, 1, 1, 2, 0, 1, 2, 1.
We copy the first two rows.
And then the third one will be 0, 1, k minus 3, and 7 minus 3
times 2, 7 minus 6, 1.
We have essentially one last row operation to perform.
Let me do it here.
So we'll subtract the second row from the third one.
And we'll get 1, 1, 1, 2, 0, 1, 2, 1, 0, 0, k
minus 5, and then 0.
And this was achieved by the elementary matrix E, 3, 2,
which was 1, 1, 1, and then negative 1.
Because we multiplied the second row by negative 1 and
added it to the third one.
So we got two matrix, which is upper triangular.
And we want to figure out for which value of the parameter k
is this matrix a full rank.
This is a pivot.
And we want this one to be a pivot as well.
And that happens when k minus 5 is not 0.
So when k is different from 5, the matrix A is a full rank.
And therefore the system Ax equals to 2, 3, 7
has a unique solution.
Now part B. For which k do we have
infinitely many solutions?
So when are we in such a situation?
We are in such a situation when the null space of the
matrix A is nontrivial.
So the null space will be nontrivial when this k minus 5
number here, which is what's the pivot in the
first case, is 0.
So k minus 5 equals to 0.
You see there's a little caveat here.
When k is equal to 5, we get the third row of the augmented
matrix, 0, 0, 0, equal to 0.
This means that the matrix is actually consistent, and we
indeed have a solution But if this entry were non 0, then we
would get no solutions.
Now off to part C. We want to compute the LU decomposition.
Well, we already got what the matrix U is through performing
row operations.
It's this guy here.
Let me write it down--
1, 1, 1, 0, 1, 2, 0, 0, k minus 5.
And when k is equal to 4, this entry's negative 1.
And now what about the matrix L?
Well, how did we get to this U?
We had the matrix A. We got the matrix U. And what we did
was first we applied E, 2, 1.
Then we applied E, 3, 1 and then E, 3, 2.
We get A by inverting this equation.
So it's going to be E, 2, 1 inverse E, 3, 1 inverse, E, 3,
2 inverse times U. And this is our matrix L. And we know it's
fairly easy to invert these elementary matrices E. We flip
the signs of the diagonal entries.
I'm not going to write down again the
inverses of these guys.
I'm just going to write the product of the inverses.
And that's also very easy to compute.
Because one, we invert the signs.
We just send the numbers and their corresponding entry of
L. I mean the following thing.
So minus 1 becomes a 1.
And it comes here, it's respective entry.
For E, 3, 1, we flip the sign, first 3, and
we plug it in here.
And for E, 3, 2, we flip the sign of this guy.
It comes 1, and we plug it in here.
So give me a few moments to erase the board, and then I'll
do part four.
We're back, and we're going to do part D now.
So we need to find the complete solution of the
system for all k.
And we solved it for k equal to 5.
The system had many, many solutions.
And for k not equal to 5, it had only one.
So first let's look at the case k not equal to 5 when the
matrix A was invertible.
It's not hard to see what the solution of
the system then is.
It's going to be--let me just write it down.
So when k is not equal to 5, this was non 0.
Therefore, x3 needs to be 0.
When x3 is 0, we have x2 plus 2 times x3.
So x2 plus 0 equals 1.
Therefore x2 is 1.
And then we go back to the first row.
We have x1 plus x2 plus x3 equals 2.
So x1 plus 1 plus 0 equals 2.
So x1 plus 1 equals 2.
And therefore x1 is 1.
Good.
Now what about k equalled to 5?
Then we see that x3 is a free variable.
So the solution will be x1, x2, x3.
x3 can be any number c.
From the second row, we'll get the value of x2.
It's 1 minus 2 times x3.
So 1 minus 2 times c.
And x1 is 2 minus x2 minus x3.
So let me rewrite this.
It's 2 minus x2 is 1 minus-- aah, chalk--
2c minus c, 1 minus 2c, c.
So we'll decompose this vector here into a component which is
independent of c and a component which is
a multiple of c.
So this is 2 minus 1.
It should be 1, 1, 0 plus c times--
we'll have two c minus c c, so c times 1.
Here we'll have minus 2c so negative 2.
And here we'll have 1.
And thus we get the particular solution for the system and
the special solution for the system.
We're kind of done here.
If you're at an exam, you should immediately start the
next problem.
Good luck, and Ill see you later.