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| == Inverse Matrices == An '''inverse matrix''' is denoted as A^-1^. If a matrix is multiplied by its inverse matrix, it returns the identity matrix. If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. For a square matrix A, the left inverse is the same as the right inverse. AA^-1^ = A^-1^A = I |
Special Matrices
Identity Matrix
The identity matrix multiplied by matrix A returns matrix A.
This matrix is simply a diagonal line of 1s in a matrix of 0s.
┌ ┐ │ 1 0 0│ │ 0 1 0│ │ 0 0 1│ └ ┘
Permutation Matrix
The permutation matrix multiplied by matrix A returns matrix C which is a mirrored transformation of A.
This matrix is simply a diagonal line of 1s in a matrix of 0s, but going the opposite direction as compared to an identity matrix
┌ ┐ │ 0 0 1│ │ 0 1 0│ │ 1 0 0│ └ ┘
Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order.
┌ ┐┌ ┐ ┌ ┐ │ 0 1│| 1 2| | 3 4| │ 1 0│| 3 4|=| 1 2| └ ┘└ ┘ └ ┘ ┌ ┐┌ ┐ ┌ ┐ | 1 2|│ 0 1│ | 2 1| | 3 4|│ 1 0│=| 4 3| └ ┘└ ┘ └ ┘
Inverse Matrices
An inverse matrix is denoted as A-1. If a matrix is multiplied by its inverse matrix, it returns the identity matrix.
If A-1 exists, then A is invertible and non-singular. Not all matrices are invertible.
For a square matrix A, the left inverse is the same as the right inverse. AA-1 = A-1A = I