In Exercises 19-28, find A^-1 by forming [A

Key Concepts

Matrix Inversion

Matrix inversion is a process used to find a matrix, known as the inverse, which when multiplied by the original matrix, yields the identity matrix. It is fundamental in solving systems of linear equations, determining linear transformations, and various areas in applied mathematics.

Augmented Matrix Method

The augmented matrix method involves setting up a combined matrix that consists of the original matrix and the identity matrix side-by-side. Through a series of row operations, the original matrix is transformed into the identity matrix, and correspondingly, the identity matrix is transformed into the inverse of the original matrix.

Elementary Row Operations

Elementary row operations are the basic maneuvers used to manipulate the rows of a matrix. They include row swapping, multiplying a row by a nonzero constant, and adding a multiple of one row to another. These operations are crucial for systematically transforming a matrix into row echelon form or the identity matrix.

Identity Matrix

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix algebra, meaning that any matrix multiplied by the identity matrix remains unchanged. It is the target form when finding the inverse of a matrix using augmented matrices.

Inverse Matrix Verification

After calculating the inverse of a matrix, it is important to verify that the solution is correct. This is done by confirming that the product of the original matrix and its computed inverse, in both orders, results in the identity matrix, thereby ensuring the inverse is properly determined.

Transcript

00:01 Okay, so in this problem, we're going to try to find a inverse using a row reduction algorithm.

00:06 So we're going to start with a on the left side here, as we see it in the problem, and the identity matrix on this side.

00:13 And we're going to row reduce a on this side, and as we row reduce a, we're actually going to make a inverse on the right side.

00:22 So let's just row reduce this and see what we get.

00:28 So to row reduce, we're going to first try to get rid of everything below the leading one in this first column here.

00:37 So what i'm going to do is i'm going to leave the first row intact because we aren't going to make any changes to it quite yet.

00:47 But we're going to use it to basically make both of these things here zero.

00:53 So we already have a zero in this spot here, which is nice.

00:57 So we can just leave that as is.

01:01 But we need to get rid of this two here.

01:04 So to do that, we're going to take everything in this row, and we're going to subtract two of these rows.

01:12 So what does that mean? so in this first spot here, it's going to be two minus two.

01:20 It's going to be two minus two of these.

01:24 So two one.

01:25 So we're going to get zero here.

01:26 And it's going to be 3 minus 2 times negative 1.

01:32 So it's going to be 3 plus 2.

01:34 So we're going to get 5 here.

01:37 Then it's going to be 0 minus 2 times 1, which is negative 2.

01:43 And it's going to be 0 minus 2 times 1, which is negative 2.

01:48 And it's going to be 0 minus 2 times 0, which is 0.

01:51 And then it's going to be 1 minus 2 times 0.

01:57 Which is one.

01:59 So in the first case, we just took the third row and subtracted out two of the first row.

02:05 So now we have zeros in these spots.

02:08 We want to get a zero now in every spot below this entry right here.

02:17 So before we do that, i'm going to take this second row and i'm going to divide it in half so that we can get a leading one.

02:28 In this spot right here.

02:30 So dividing each of these terms by two is going to give us one here.

02:38 We're going to get negative one half here.

02:44 All the zeros are still going to be left in place.

02:47 But this one, we're dividing it by two, so it's going to become just a regular one half.

02:53 So we just wanted to get a leading one in the position right above the five so that now we can cancel out the five.

02:58 And the process looks exactly the same.

03:02 So we're going to take, we're going to keep everything else constant here because we're not changing anything except for the third row.

03:14 1 -1 -0 -1 -5 -1 -0 -0 -1 -5.

03:29 And then we're going to take the third row minus 5 to the second row.

03:34 So it's going to be 0 minus 5 -0s, which is 0.

03:37 0 still.

03:38 It's going to be 5 minus 5 minus 5 1s up here.

03:44 So it's going to be 0 here, which is what we were trying to do.

03:48 It's going to be 2 minus 5 of the halves, which is going to give us just positive 1 1⁄2.

03:57 It's going to be negative 2 minus 5 zeros, which is still negative 2.

04:02 It's going to be 0 minus 5 1 halves.

04:06 So we're going to minus 5 halves here.

04:09 It's going to be 1 minus 5 zeros, which is still 1 here.

04:14 And so we're going to do another similar thing.

04:17 We want this to be a 1 right here, this bottom entry.

04:20 So we're just going to multiply everything in the bottom row by 2.

04:23 So 1⁄2 is going to be 1.

04:27 Negative 2 times 2 is going to be negative 4.

04:33 Negative 5 halves times 2.

04:35 It's going to be negative 5.

04:38 And then 1 times 2 is going to be 2.

04:44 All right, so we've almost done our row reduction.

04:47 We've gotten all these zeros below the leading ones, which is what we wanted.

04:52 Now we need to go back up the other way, get rid of this negative 1⁄2, this 1, this 1, this negative 1.

04:58 So to do that, we're going to look, it's going to look very similar to what we just did.

05:04 We're just going to go in the opposite direction.

05:08 So we're going to first take this, this third one right here and use it to get rid of this negative one half and this one up here.

05:20 So what's that going to look like? well, we're going to keep this last row intact.

05:26 Negative four, negative five.

05:32 And to get rid of this negative one half first is going to mean that we are going to take, we're going to take two row twos and add that to a row.

05:47 A row three.

05:49 So it's going to be zero.

05:51 So for this first entry, it's going to be zero plus, or two zeros plus zero, which is zero.

05:57 It's going to be, yeah, well, let's do it the opposite way.

06:06 So instead of taking two of row twos, we'll just, we'll keep, we'll take two of row, or a half of row three and add that to the first one.

06:18 So this first term is still going to be the same, but now it'd be like one half times zero plus one, which is going to keep the one.

06:26 I just wanted to keep this one in place, and i didn't want to change it.

06:29 If we did it the other way, it would have changed it.

06:31 Then we can do one half times one, which is a half, plus negative a half, which is zero.

06:38 One half times negative four plus zero, which is negative two.

06:43 One half times negative five, which is going to give us negative five halves, plus one half is going to be negative four halves, which is negative two.

06:55 One half times two, which is negative one, plus zero, which is one.

07:02 So that's going to be that row right there.

07:04 Now we need to get rid of this one up here.

07:06 So we're just going to take that first row and subtract one of the last rows...

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