The inverse of a matrix A is written as A-1. When a matrix is multiplied by its inverse, or when the inverse is multiplied by the original matrix, the resultant matrix should have 1's in its main diagonal, while every other entry is 0.
An easy method of finding the inverse of a matrix is to juxtapose a matrix in the desired format to it, and then to perform the basic row operations until the matrix portion on the left looks like the desired matrix. The matrix portion on the right will be the inverse. This might sound a bit confusing, so I will explain the process with a simple 2 x 2 matrix I found on the Internet (by the way, not all matrices are invertible, so some might not have an inverse at all).
Here are the steps to solving this matrix:
- Add the first row multiplied by -3 to the second row.
- Divide the second row by -2.
- Add the second row multiplied by -2 to the first row.
Now, we have determined that the matrix on the right side of the vertical line is the inverse of the original matrix.
You can always verify that you have correctly determined the inverse matrix by either multiplying the original matrix by its inverse or vice-versa; if the resultant matrix consists entirely of 0's except for 1's in the main diagonal, then the inverse is correct!
You can always verify that you have correctly determined the inverse matrix by either multiplying the original matrix by its inverse or vice-versa; if the resultant matrix consists entirely of 0's except for 1's in the main diagonal, then the inverse is correct!
Great post! I like how you clearly walked through the steps of solving for the matrix' inverse. I also like how you explained how to verify these problems by multiplying the original matrix with the inverse to see if it results in a matrix in reduced row-echelon form.
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