00:01
We want to solve this system of three equations by inverting the coefficient matrix.
00:12
First of all, let's turn this system of equations into a linear matrix system.
00:23
So that's writing this system in terms of vectors and a matrix, that is the coefficient matrix.
00:32
The coefficient matrix is the matrix we get just from all the coefficients.
00:40
In the exact same order as they appeared in the three equations.
00:55
And then this whole left side is just the coefficient matrix multiplied by the unknown vector, the vector of variables.
01:20
And so we have this system.
01:22
Now we want to find the inverse of this coefficient matrix.
01:30
To do this, we form an augmented matrix with the matrix we're inverting on the left and an identity matrix on the right, and then do row -tranex.
01:39
Transformations until we get the identity matrix on the left, and thus the inverse matrix on the right.
01:49
Note that this is not the most efficient way of solving this system, since we could just start with an augmented matrix with the unknown vector on the right side, 676, and do row transformations until we get the identity matrix on the left.
02:09
That will just give us the solution vector on the right.
02:13
However, this question specifically asks us to invert this matrix, so that's what we're going to do.
02:22
To do this first, we want to get rid of these, or we want to form the identity matrix on the left, right? so, first let's get just one non -zero entry in the first column.
02:39
Let's do this one, since it's already a one, and we want only ones in the identity matrix.
02:45
So we have to cancel out these two, and then later on we'll cancel out non -zero entries in the second column and in the third column.
03:00
So to cancel out these two non -zero entries, we subtract twice the first row from the second to get this new second row, and then we subtract three times the first row from the third row to get this...