Find the inverse of each matrix. [ -1 -1 -1 3 3 4 0 1 0 ]

Find the inverse of each matrix. [ -1 -1 -1 ...

Key Concepts

Adjugate Matrix and Cofactor Method

The adjugate matrix, which is the transpose of the cofactor matrix, is used in one method of finding the inverse of a matrix. When the determinant is non-zero, the inverse of a matrix can be computed by dividing the adjugate matrix by the determinant. This method systematically uses cofactors to construct the inverse, making it a key concept in theoretical treatments of matrix inversion.

Gaussian Elimination and Elementary Row Operations

Gaussian elimination is an algorithmic method used to solve linear systems and compute the inverse of a matrix by transforming it into reduced row echelon form. The process involves performing a series of elementary row operations on the original matrix and the identity matrix concurrently, eventually yielding the inverse matrix, provided the matrix is invertible.

Matrix Inverse

The matrix inverse is a fundamental concept in linear algebra where, for a square matrix A, an inverse matrix A?¹ exists provided A is non-singular, meaning its determinant is non-zero. The product of a matrix and its inverse (in either order) yields the identity matrix, a property that is central to solving linear equations and understanding linear transformations.

Determinant

The determinant is a scalar value associated with a square matrix that provides important information about the matrix, including whether it is invertible. A non-zero determinant indicates that the matrix has an inverse, while a zero determinant implies that the matrix is singular. Calculating the determinant is a crucial step when trying to obtain the inverse of a matrix.

Transcript

00:01 Okay, so for this matrix, we want to find the inverse, this 3x3 matrix will call a.

00:05 We're going to use this idea that the inverse is equal to 1 over the determinant times the adjunct of a.

00:15 And so if we want to find the determinant, we have multiple methods that we can use to do that.

00:20 If you're unfamiliar with that, check out some of the previous videos.

00:24 But to do that here, we would get plus 0 minus 3 minus 3.

00:42 0.

00:43 Okay.

00:45 So i was using the row method where we pick a row and we take the coefficients of that row, times the determinants of the matrix that is formed by removing that row and column.

01:02 But there are multiple ways that you could find the determinant of matrix.

01:05 So this is going to be 4 minus 3, which is 1.

01:09 So our 1 over our determinant is just going to be 1.

01:12 So then all we actually really need defined is this adjunct of a, which is going to be itself a three by three matrix.

01:22 And how we do that is a little bit convoluted.

01:26 We're going to copy down our matrix numbers.

01:31 And then we're going to append the first two.

01:36 Whoops, and i did that incorrectly.

01:39 This should be one, zero.

01:41 And then we're going to copy down the first two columns onto the end of it.

01:46 So these two columns, we just copy them onto that.

01:49 And after we've done that, we're going to copy the first two rows onto the bottom of it.

01:56 So we're just extending our matrix like this.

02:02 And then we can ignore the first row and the first column...

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