Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. [ 2 4 1

Finding the Inverse of a Matrix Find the inverse of the...

Key Concepts

Adjugate/Cofactor Method

The adjugate or cofactor method is an alternative approach for computing the inverse of a matrix. It involves calculating the matrix of cofactors, transposing it to get the adjugate, and then dividing by the determinant. This method provides a formulaic route to inversion especially in theoretical contexts.

Elementary Row Operations

Elementary row operations, which include row swapping, scaling, and row addition, are techniques used to manipulate matrices. These operations are essential in methods like Gauss-Jordan elimination to reduce a matrix to its reduced row echelon form for finding the inverse or solving systems of equations.

Matrix Inversion

Matrix inversion is a process of finding another matrix, often denoted as the inverse, such that when it multiplies the original matrix, the result is the identity matrix. This concept is fundamental in solving systems of linear equations and understanding linear transformations within linear algebra.

Determinant

The determinant is a scalar value calculated from a square matrix that provides key insights about the matrix properties, such as whether the matrix is invertible or singular. A non-zero determinant implies that the matrix has an inverse, whereas a zero determinant indicates that it does not.

Transcript

00:01 Let's say we want to take the 3x3 -4 -1 -1 -1 -1 -1 -14 -0 and find its inverse.

00:08 We're going to do this by extending it to a 3 -6 matrix with the 3 -by -3 identity matrix, and we'll do some row operations in order to reduce the left side to the identity matrix, and then what's the left on the right side will be our inverse.

00:28 So let's do a few things at once here.

00:37 Let's take half of the first row.

00:46 And if we do that, we've got half of the entire first row would be one, two, one, one half zero, and we've got the other rows remaining the same.

01:11 Now let's add what if we add actually already out of the gate made a slight mistake half of one is one half here as well okay so now what if we were to take how about we add rows one in two together and replace that with row two and also add rows one and three together and replace that with row three.

02:02 So now we're going to leave row one the same.

02:11 Row two is adding rows one and two together.

02:15 So we'll have 0, 3, negative 1 half, 1 half, 1 .0 for row 2 and for row 3, that's going to be formed by adding 1 and 3 together.

02:32 So that would be 1 plus 1, 2 plus 4, 1 half plus 0, 1 half plus 0.

02:37 1 1 half plus 0, 0 plus 0, and 0 plus 1.

02:56 I actually see something different.

02:58 I'm changing my mind on the row operation here.

03:03 Let's replace row 3 with the difference of row 3 in row 1.

03:13 So it would be 1 minus 1, 4 minus 2, 0 minus 1 1⁄2, 0 minus 1⁄2, 0 minus 1⁄2, 0 minus 1⁄2, 0 minus 0 and 1 minus 0.

03:29 Now, moving forward, we can say, let's multiply the second row by one -third.

03:52 If we multiply the second row by one -third, we'll have a second row of 0 -1 -6 -1 -6 -1 -3rd, and the other rows will remain the same...

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