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

Key Concepts

Matrix Invertibility

Matrix invertibility refers to the property of a square matrix having an inverse such that when the matrix is multiplied by its inverse, the result is the identity matrix. This concept is central to solving linear systems and understanding linear transformations.

Determinant

The determinant is a scalar value derived from a square matrix that provides important information about the matrix, such as whether it is invertible. A nonzero determinant indicates that an inverse exists, while a determinant of zero means the matrix does not have an inverse.

Gaussian Elimination

Gaussian elimination is a systematic method for performing row operations to reduce a matrix to its row echelon form or reduced row echelon form. This technique can be used to find the inverse of a matrix by augmenting it with the identity matrix and then row reducing until the original matrix is transformed into the identity, resulting in the augmented part being the inverse.

Adjugate and Cofactor Method

The adjugate and cofactor method involves computing the matrix of cofactors, transposing it to form the adjugate matrix, and then dividing by the determinant of the original matrix to obtain the inverse. This method is particularly useful for theoretical analyses and smaller matrices.

Transcript

00:01 So we're given the following matrix and we want to find its multiplicative inverse if it exists.

00:06 So in order to determine the multiplicative inverse, we're going to rewrite this as the augmented matrix between this one and the identity matrix for a 3 by 3.

00:16 So the identity matrix for a 3 by 3 is just 1 -0 -0 -0 -0 -0 -0.

00:26 So if we rewrite this as the augmented matrix, we'll have 1 .7.

00:32 06, negative 2, 17, 3 ,02, 100, 010, 010, 0 .01.

00:45 And so the purpose of writing this matrix into this form is so that we can turn a into i using row operations, so that i will become a inverse.

00:58 So this is a, matrix a, and this is the identity matrix.

01:04 So let's start by using row operations to turn a into i.

01:11 So the first one will be to multiply the first row by negative 3 and added to the third row.

01:20 So if we do this, we'll have 1, 0, 0, 6, negative 3 times 1 is negative 3.

01:28 Oh, sorry, added to the third row, so the second row is the same, 1 and 7.

01:33 So this becomes 0, this stays as 0.

01:37 And our third row, so negative 3 times 6 is negative 18, negative 18 plus 2 is negative 16.

01:46 So the first two rows on the right hand side stay the same.

01:50 Our third row just becomes negative 3, 0 and 1.

01:56 The next row operation will be to multiply row 3 by negative 1 over 16.

02:06 So 1 .06 -10 for the first row, negative 2, 1 -7010 for the first row, negative 2, 1 -7 -0 for the 1.

02:15 Second row 0 -0 -1 negative 3 over 16 0 and 1 over 16 sorry negative 1 this is positive 3 over 16 now we're going to multiply row row 3 by negative 6 so negative 6 times row 3 we're going to add that to row 1 so the first the third row and second row stay the same 3 over 16 0 1 over 16 0 1 over 16 010...

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