Find the inverse of the given matrix. [ 1 -2 3 0 0 1 0 2 1 4 -2 3

Find the inverse of the given matrix. [ 1 -2 3 0 ...

Key Concepts

Matrix Inversion

Matrix inversion refers to the process of finding a matrix A?¹ such that when it is multiplied by the original square matrix A, the result is the identity matrix. This concept is central in solving linear systems, understanding linear transformations, and in various computational applications.

Determinant and Invertibility

A key concept in determining whether a matrix has an inverse is the determinant. For a square matrix, a nonzero determinant indicates that the matrix is invertible, while a zero determinant means that the matrix does not have an inverse.

Augmented Matrix Method

One common method to find the inverse of a matrix involves creating an augmented matrix by appending the identity matrix to the original matrix. Through a series of row operations, the original matrix is reduced to the identity matrix, and the transformations applied to the identity matrix yield the inverse of the original matrix.

Elementary Row Operations

Elementary row operations—which include row switching, scaling rows by a nonzero constant, and adding multiples of one row to another—are the fundamental steps used to perform Gaussian elimination on augmented matrices. These operations maintain the equivalence of the linear system and are essential in the process of matrix inversion.

Transcript

00:01 Number 433.

00:04 We have a 4x4 matrix here, and we want to find the inverse.

00:10 Okay.

00:11 So the way to do that is to write the matrix, and then on the right -hand side, we write the identity matrix of the same size.

00:33 So all zeros, except for ones in the main diagonal.

00:41 And that's the identity matrix.

00:43 We want to perform row operations until we get the identity matrix on the left.

00:52 So starting in the first column, we have a one in the upper left, which is what we want, and we want the rest of the numbers to be zero.

01:03 So i am going to subtract row three minus row one, and i am going to subtract, i'm going to subtract, row 4 plus 5 times row 1 okay so when we do row 3 minus row 1 we'll get a 1 minus 1 and we'll get a 0 0 here and when we do row 4 plus 5 row 1 we'll get negative 5 plus 5 and we'll get a 0 here as well okay so rewriting the first two rows okay now we're subtracting each element from row from row three.

02:26 I mean, we're subtracting each element from row one, in row one from row three.

02:34 So one minus one is zero.

02:38 Four minus negative two is six and so on.

02:57 Okay, now we're adding row four plus five row one.

03:01 Negative five plus five times one is zero.

03:05 0 plus 5 times negative 2 is negative 10 and so on.

03:30 Okay, our first column is complete.

03:34 Now on the second column, we want to get a 1 in this position, which we have already, and we want to get zeros here, here, and here.

03:48 So i am going to multiply row 2 times 2 and add it to this.

03:55 This to get zero.

03:59 And negative six times row two and added to this to get zero.

04:04 10 times row two and added to this to get zero.

04:08 Okay.

04:10 So we're going to do two row two times row two plus row one, negative six row two plus row three, and ten row two plus row four.

04:38 Okay, so first row, two times zero plus one, two times one plus negative two, and so on.

05:17 Rewriting the second row, third row, negative six times zero plus zero, negative six times one plus six, and continue in the same fashion.

05:57 Okay so now finally 10 times row 2 plus row 4 well here we just have 0 10 times 1 plus negative 10 is just 0 and continue in the same fashion okay so now we want to work on column 3 we want a one right here so i am going to i'm going to multiply row three times one over five or should i say negative one over five negative one over five times row three okay and just to make it simple i'm going to erase this instead of rewriting everything okay so here we'll have a one here we'll have a one okay and um multiply everything oops, what happened? multiply everything by negative one -fifth.

07:51 So you should get nine -fifths here.

07:58 Should get nine -fifths right here.

08:02 One -fifth, six -fifths, negative one -fifth, and zero.

08:16 Okay.

08:18 So now we have a new row three.

08:27 Okay.

08:28 And we want a zero in place of the three.

08:31 A zero in place of the 16.

08:35 So i am going to do negative three times row three plus row one.

08:50 And i'm going to do negative 16 times row three plus row four.

09:02 All right.

09:06 So let's see.