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

step 1 Okay, so here we're given our matrix A. So we write down a matrix A and then a vertical line and

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

Key Concepts

Inverse Matrix

An inverse matrix of a given square matrix is another matrix that, when multiplied with the original, yields the identity matrix. It acts as the ‘reciprocal’ for matrices and is essential in solving linear systems, where finding the inverse allows one to derive solutions through matrix multiplication.

Non-Singular Matrix

A non-singular matrix is a square matrix that possesses an inverse, which is guaranteed when its determinant is non-zero. This property implies that the matrix has full rank and that the associated linear system has a unique solution.

Elementary Row Operations

Elementary row operations, including row switching, scaling rows by non-zero constants, and adding multiples of one row to another, are fundamental procedures used in matrix manipulation. These operations preserve the solutions of the matrix system and are crucial for transforming a matrix into reduced forms during processes like finding an inverse.

Gauss-Jordan Elimination

Gauss-Jordan elimination is a systematic method for reducing a matrix to its reduced row echelon form using elementary row operations. It is frequently used to compute the inverse of a matrix by augmenting the matrix with the identity matrix and then applying these operations until the original matrix is transformed into the identity matrix, resulting in the inverse appearing in the augmented portion of the matrix.

Transcript

00:01 Okay, so here we're given our matrix a.

00:03 So we write down a matrix a and then a vertical line and then we put down the three by three identity matrix.

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

00:13 Now we want to perform just elementary row operations, putting this in reduced row echelon form, where we end up converting the left hand side to the identity.

00:23 And then what we have on the right hand side is going to be our inverse matrix.

00:27 So we have a 1 already in our first spot.

00:30 We want to have zeros beneath it.

00:32 So let's do for row 2, let's do row 2 minus 3 times row 1.

00:44 And then for row 3, let's do row 3 minus 3 times row 1 into row 3.

00:53 That is then going to put us at, well, we have 1, 1, but then we have 01, 1, but then we have 0 .0.

00:58 Is beneath it.

01:00 And then we have negative 1, 4, negative 1, negative 4, and negative 2, negative 1.

01:08 Negative 1.

01:09 And then on the right hand side, now we have 1 -0, negative 3, 10, 1 .1.

01:23 So next, well, we got to have our next pivot spot here.

01:29 And then second row, second column needs to be a one.