Find the multiplicative inverse of each matrix, if it exists...

Key Concepts

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It plays a crucial role in understanding the properties of the matrix, including invertibility. A non-zero determinant indicates that the matrix is non-singular (invertible), while a zero determinant means that the matrix is singular and does not have an inverse.

Matrix Inversion

Matrix inversion is the process of finding another matrix which, when multiplied with the original matrix, yields the identity matrix. This concept is fundamental in solving systems of linear equations, understanding linear transformations, and many other applications in linear algebra. The existence of an inverse is restricted to square matrices and is contingent on the matrix being non-singular.

Adjugate Matrix

The adjugate matrix is the transpose of the cofactor matrix of a given square matrix. It is used in the formula for finding the inverse of a matrix, where the inverse is given by the adjugate divided by the determinant of the original matrix. This method is particularly useful for theoretical analysis and symbolic computation of inverses.

Invertibility Condition

Invertibility condition refers to necessary and sufficient criteria that determine whether a matrix has an inverse. For a square matrix, it must have a non-zero determinant and full rank. Understanding these conditions is essential when assessing whether or not an inverse exists, and it also highlights the important relationships between linear independence, matrix rank, and the ability to define a unique inverse.

Transcript

00:01 Problem 409, they want us to find the inverse of this three by three matrix here.

00:06 The way we do that is to rewrite the matrix and then write the identity matrix on the right.

00:29 Okay.

00:31 And we're going to perform row operations until we get the identity matrix on the left.

00:52 Okay.

00:53 So the first step i would like to get, i would like to get rid of the common denominator here.

01:01 And i would like to get all ones on the left column.

01:05 So i'm going to multiply row one by two.

01:14 I'm going to multiply row two by three.

01:19 And i'm going to multiply row three by six.

01:38 Okay, row two, each term multiplied by three.

01:42 And row three multiply each term by six.

02:13 Okay.

02:15 Now, i want to get, i want to keep this one here, and i want to get zeros right here.

02:24 So i'm going to do row two minus row one and put it in row two, and i'm going to do row three minus row one and put it in row three.

02:43 So rewriting row one.

02:56 Okay, in row two minus row one, we're subtracting each term in row two.

03:02 We're subtracting each term in row one from each term in row two.

03:06 So basically we're subtracting up.

03:46 Okay, so just real quick, row two minus row one, we did one minus one.

03:51 We got zero.

03:52 We did three -fourths minus one.

03:54 We got negative one -fourth.

03:56 Three -fifths minus one.

03:58 We got negative two -fifths.

03:59 So on.

04:01 And for row three, one minus one, we got zero.

04:05 Six -sevenths minus one, negative one -sevenths, six -eighths minus one, negative two -eighths, and so on.

04:14 All right.

04:16 The next step, i would like to get ones right here in the middle column.

04:24 So i'm going to multiply row two by negative four, and i'm going to multiply row three by negative seven.

04:44 All right, so we have to multiply each term in the row.

04:50 And rewrite the first row.

04:58 Okay, each term by negative four.

05:02 We get eight fifths.

05:12 And row three, multiply each term by negative seven.

05:29 All right.

05:36 So let me see, let me see...

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