Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. [ 4 2 3

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

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding another matrix such that when it is multiplied by the original matrix, the result is the identity matrix. The inverse exists only for square matrices that are nonsingular, meaning they have full rank. This concept is fundamental in solving linear systems and in many applications across mathematics and engineering.

Determinant

The determinant is a scalar value that is computed from the elements of a square matrix. It plays a critical role in determining whether an inverse exists; specifically, a matrix is invertible only if its determinant is not zero. In addition, the determinant is useful for understanding various properties of the matrix, such as scaling factors in linear transformations.

Adjugate Matrix

The adjugate matrix, or adjoint matrix, is formed by taking the transpose of the cofactor matrix of a given square matrix. It is used in the formula for calculating the inverse of a matrix, where the inverse is equal to the adjugate matrix divided by the determinant. This method is particularly important when dealing with symbolic matrices or smaller matrices where the manual computation is feasible.

Transcript

00:01 So to find the inverse of this matrix, the first step is to create a matrix of minors, meaning that for each element of the matrix, we ignore the values on the current row and column, and then we calculate the determinant of the remaining values.

00:22 For example, assuming we have a matrix that looks like this.

00:34 And let's say we're working on a.

00:37 So we're just going to ignore the row and the column and calculate the determinant for the remaining value.

00:47 So in this case, it will be ei minus fh.

00:51 And you do it for the rest of the values.

00:55 And back to our problem.

01:00 In our case, we would get 3 ,201 for 4, and then 3211, and then 321, and then 3, 3 ,0, and then 2 -3 -0 -1 and then 4 3 1 1 210 and then 2 3 3 2 and then 4 3 2 and then if we calculate each of those we would end up gain 3 1 negative 3 2, 1, negative 2, and negative 2, and negative 5, negative 1, and 6.

02:12 And the next step would be the matrix of cofactors.

02:24 And all we have to do is just apply the checkerboard of minuses to the matrix of minors.

02:30 And it's going to look like this.

02:33 This is our matrix of minors, and then we just multiply that by the checkerboard meaning.

02:44 Meaning and then we would get 3 negative 1 negative 3 then negative 2 1 2 and negative 5 negative 1 then 6 and we're done with the step and the next step is it's called the adjugate, and we're going to transpose all the elements from the previous matrix by swapping the positions over the diagonal.

03:43 And what i mean by that, here is our matrix of co -factor...

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