In Problems 31-40, each matrix is non singular. Find the inverse of each matrix. [ 3 3

In Problems 31-40, each matrix is non singular. Find the...

Key Concepts

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithm used to reduce a square matrix to its reduced row echelon form through a series of elementary row operations. When these operations are applied to the identity matrix concurrently, the resulting matrix becomes the inverse of the original matrix.

Adjugate (Cofactor) Method

The adjugate method involves calculating the matrix of cofactors for each element, transposing that matrix to obtain the adjugate, and then dividing by the determinant of the original matrix. This process results in the inverse of the original matrix when the determinant is nonzero.

Matrix Inversion

Matrix inversion is the process of finding a matrix, often denoted as A?¹, such that when it is multiplied by the original matrix A, the result is the identity matrix. This concept is fundamental in solving systems of linear equations and is only applicable if A is a square, nonsingular matrix.

Nonsingular Matrix

A nonsingular matrix, or an invertible matrix, is one that has an inverse. This property is determined by ensuring that the matrix’s determinant is nonzero, indicating that its rows and columns are linearly independent and that the matrix does not collapse the n-dimensional space into a lower dimension.

Matrix Determinant

The determinant is a scalar value computed from the elements of a square matrix that indicates whether the matrix has an inverse. A nonzero determinant means the matrix is nonsingular (and hence invertible), while a determinant of zero shows that the matrix is singular and cannot be inverted.

Transcript

00:01 Okay, so given our matrix a, then we write down our matrix a, and then we put a vertical line, and then we put the three -by -three identity matrix.

00:09 So that's just the matrix with ones down the main diagonal and zeros everywhere else.

00:13 Now, you want to use just elementary row operations, putting this in reduced row echelon form, where we put the identity we want on the left -hand side of this vertical bar.

00:24 And then what we end up, what we end up with on the right -hand side of that vertical bar is going to be our inverse.

00:30 Matrix.

00:31 So we want to get this in the left -hand side.

00:34 We want the identity.

00:36 So you want to have a one, right, is our leading spot here.

00:40 So right now we have a three.

00:41 Let's go ahead and multiply row one by one -third.

00:44 And then we want to have zeros beneath that.

00:48 So what we want to do is first we want to do for row one.

00:51 We want to do one -third times row one.

00:54 So one -third times row one into row one.

00:57 And then for row two, let's do row two.

01:02 Let's do row two minus one -third times row one.

01:09 And then for row three, let's do row three minus two -thirds times row one.

01:18 That's going to then put a one in our leading spot and with zeros beneath it.

01:25 So doing that, we end up with this here.

01:29 We end up with our first row becomes one, one, one third, and then we have zero, one, two thirds, and then we have zero, negative three, one third.

01:43 And then we have a vertical bar in the right hand side...

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