Find the inverse of each matrix. [ 0 1 0 3 5 2 1 2 1 ]

Find the inverse of each matrix. [ 0 1 0 3 ...

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix. This concept is critical in linear algebra as it is used to solve systems of linear equations, analyze linear transformations, and compute solutions in various applied mathematics fields.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible—only matrices with a nonzero determinant have an inverse—and it plays a key role in understanding the properties of linear transformations.

Adjugate (Adjoint) Matrix

The adjugate, or adjoint, of a matrix is the transpose of its cofactor matrix. It is used in the formula for the inverse of a matrix, specifically when the inverse is expressed as the adjugate divided by the determinant, making it an essential concept in the computation of matrix inverses.

Elementary Row Operations and Gauss-Jordan Elimination

These are methods used to systematically manipulate a matrix into a desired form, such as the reduced row echelon form or the identity matrix. In the context of finding an inverse, the Gauss-Jordan elimination process transforms the augmented matrix (original matrix and identity matrix) such that the original matrix becomes the identity matrix, and the transformation applied to the identity matrix yields the inverse.

Transcript

00:01 Okay, so for this question, we're to ask to find the inverse of a 3x3 matrix here.

00:06 We'll call it a.

00:07 I'm proposing that we use this formula that says that a inverse is equal to 1 over the determinant of a times the adjunct of a.

00:24 So how we're going to find the determinant is by the method of picking a number from a row.

00:34 We're going to use the top row here, but you can actually use this method for any row.

00:38 And so we're going to pick 0, and then we're going to get the remaining.

00:45 If we take out the row and column that 0 is in, and we find the determinant of the remaining matrix and multiply it by 0, we're going to use that portion.

00:52 So the determinant of the remaining here would be 5 minus 2, 5 minus 4 rather, but it's multiplied by 0, so it doesn't matter.

01:00 Then we subtract off 1 times 3 minus 2.

01:09 And then again, we're going to have plus 0, eliminate the row and columns.

01:14 The determinant of this matrix is going to be 6 minus 5.

01:23 And in the end, the only thing that actually stays is this middle term.

01:27 And so we get that the determinant is minus 1.

01:30 So 1 over minus 1 is just going to be minus 1.

01:34 So then we need to find out what this matrix.

01:40 Is.

01:46 And the way that we're going to do that is through a kind of convoluted method.

01:50 So we're going to write out this whole matrix.

01:53 0 .10 .352.

01:58 1, 2.

01:59 And we're going to copy down the first two columns onto the right side of it...

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