In Problems 31-40, each matrix is non singular. Find the inverse of each matrix. [ 1 -1

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

Key Concepts

Elementary Row Operations

Elementary row operations, which include row swapping, scaling, and adding multiples of rows, are used to transform a matrix into a simpler form such as the row-echelon or reduced row-echelon form. These operations are essential in methods like Gauss-Jordan elimination, which can be used to compute the inverse of a matrix.

Non-Singular Matrix

A non-singular matrix is one that has an inverse, meaning its determinant is non-zero. This property is crucial because it guarantees the existence of a unique solution to the matrix equation Ax = b for every vector b, and it is a necessary condition for many applications in linear algebra.

Matrix Inversion

Matrix inversion is the process of finding another matrix which, when multiplied by the original matrix, yields the identity matrix. It is a fundamental operation in linear algebra used in solving systems of linear equations, computing matrix functions, and various applications in engineering and science.

Determinant

The determinant of a matrix is a scalar value that provides important information about the matrix, such as whether it is singular or non-singular. A non-zero determinant indicates that the matrix is invertible and helps in understanding the matrix's behavior under linear transformations.

Adjugate Matrix

The adjugate (or classical adjoint) of a matrix is derived from its cofactors and is used in the formula for the inverse of a matrix. Specifically, the inverse of a non-singular matrix can be computed as the adjugate of the matrix divided by its determinant.

Transcript

0:00 Hello.

00:01 So here we're given our matrix a, and then we go ahead, we basically draw a vertical line and then put the 3 by 3 identity matrix.

00:09 That's i3, which is just the matrix with ones on the main diagonal and zeros everywhere else.

00:14 Now it's going to perform elementary raw operations to basically put the identity on the left hand side, and then what we end up with on the right hand side is going to be our inverse matrix.

00:25 So, well, we already have the one, right, in our first, spot here.

00:32 We want to have well, zeros all the way beneath it.

00:34 So therefore, we want to put a zero in the third row first column here.

00:40 So let's do row three be equal to row three minus two times row one.

00:52 So doing that, we then end up with on the left hand side while we end up with 1, negative 1, 1, 0, negative 2, 1.

01:03 And then we have our 0 in our spot here.

01:06 And then we have negative 5 and 2.

01:11 And then we put our vertical line.

01:15 And then we have, on the right hand side, we end up with, well, 1 -0 -0 -0 -0 and 201, and 2 -1.

01:26 And 2 -1.

01:28 Okay.

01:30 And then what do we want? well, we want to have getting a one right in our next pivot spot here and then zeros above and below.