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

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

Key Concepts

Inverse Matrix

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. This concept is central in linear algebra as it allows for the solving of systems of linear equations, transforming coordinate systems, and more. The existence of an inverse is tied to the property of a square matrix having a nonzero determinant.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and encapsulates several properties of the matrix, including whether it is invertible. A nonzero determinant ensures that the matrix has an inverse, while a determinant of zero indicates that the matrix is singular and does not have an inverse.

Gaussian Elimination

Gaussian elimination is a systematic method used to simplify matrices through the application of row operations. In the context of finding an inverse, this technique is applied to form an augmented matrix with the identity matrix and then use elementary row operations to reduce the original matrix to the identity matrix, simultaneously transforming the identity matrix into the inverse.

Adjugate Method

The adjugate method involves calculating the matrix of cofactors (or the adjugate) and dividing by the determinant to find the inverse. This method provides a formulaic approach to inversion and highlights the interplay between determinants, minors, and cofactors in establishing the inverse of a matrix.

Transcript

00:01 Okay, to find the inverse of this matrix, say we're going to use what's called the shortcut method by some people.

00:08 And so we have a formula.

00:10 The inverse of our matrix is one over the determinant of the matrix times what's called the adjunct of a.

00:18 So to find the determinant, we're going to use the method where we pick a coefficient here, say one, and then get the determinant of the matrix that's the result of eliminating that column in row that that coefficient falls in.

00:30 So we're going to get 1 times the determinant of this guy.

00:34 So we'll have 0 minus 5.

00:37 So that's just a negative 5 plus 1 times 2 plus 2.

00:50 And then we will have 0 times something, but we don't really care what it is since it's going to be 0.

00:58 And so this is the determinant of a.

01:02 And we see that we get minus 5 plus 4.

01:05 It's going to be minus 1.

01:06 So 1 over its determinant is just going to be minus 1.

01:10 So once we find this adjunct matrix a, we can just multiply it by a negative 1, and that'll be our inverse.

01:22 So how we find this matrix is by taking kind of a convoluted series of steps, we're going to rewrite out our matrix.

01:35 And then after we get this all written out, we're going to take the first and second column and add them again onto the right.

01:43 Of all of these numbers...

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