# Matrix Inverse

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Matrix Inverse () is a monadic function that returns the inverse of the simple numeric array of rank 2 or lower. It shares the glyph Quad Divide (often called Domino) with the dyadic function Matrix Divide.

## Examples

Matrix Inverse computes the ordinary inverse if the argument is a square matrix. DOMAIN ERROR is raised if the given matrix is not invertible.

M2 23 4 4 5
3 4
4 5
RM
¯5  4
4 ¯3
R+.×M
1 0
0 1
2 20
DOMAIN ERROR
2 20

When the argument is a scalar or vector, or the given matrix has more rows than columns (r>c where r c≡⍴X), Matrix Inverse computes specific forms of generalized inverse called Moore-Penrose inverse. For a scalar, the result is the reciprocal of the argument; for a vector, the result equals (+X)÷X+.×+X. For a non-square matrix, the result equals (+⍉X)(+⍉X)+.×X (where +⍉X is the conjugate transpose of X).

(2)(2J1)
0.5 0.4J¯0.2
÷2 2J1
0.5 0.4J¯0.2

(3 1)(2 1 1J2)
┌───────┬────────────────┐
0.3 0.10.2 0.1 0.1J¯0.2
└───────┴────────────────┘
{(+)÷+.×+}¨ (3 1) (2 1 1J2)
┌───────┬────────────────┐
0.3 0.10.2 0.1 0.1J¯0.2
└───────┴────────────────┘
(3 1)(2 1 1J2) +.×¨ (3 1)(2 1 1J2)
1 1

M3 21 ¯1 0J1 1 ¯1 0J1
1   ¯1
0J1  1
¯1    0J1
RM
0.5J¯0.5 0.25J¯0.25 ¯0.25J¯0.25
¯0.5J¯0.5 0.25J¯0.25 ¯0.25J¯0.25
R{(+⍉)(+⍉)+.×} M
1
R+.×M
1.0000E000J¯5.5511E¯17 0
¯2.7756E¯17J05.5511E¯17 1