# Inverse of a matrix in ℤ/pℤ: Inverse

### 5.35.6 Inverse of a matrix in ℤ/pℤ: Inverse

In Xcas mode, Inverse is simply the inert form of inverse; namely, it gives the inverse of a matrix without evaluating it. (See section Section 5.47.2.) In Maple mode, the Inverse command can additionally be used in conjunction with mod to find the inverse of a matrix whose elements are in ℤ/pℤ.

- (In Maple mode.)
  Inverse takes one argument:
  A, a matrix with elements in ℤ/pℤ.
- Det(A) returns the inverse of A.

#### Example.

Input (in Xcas mode):

Inverse([[1,2,9] mod 13,[3,10,0] mod 13,[3,11,1] mod13])

Output:

inverse

⎛
⎜
⎜
⎜
⎜
⎜
⎝

⎡
⎢
⎢
⎢
⎢
⎢
⎣

1%13

2%13

⎛ ⎝ | −4 | ⎞ ⎠ | %13

3%13

⎛ ⎝ | −3 | ⎞ ⎠ | %13

0%13

3%13

⎛ ⎝ | −2 | ⎞ ⎠ | %13

1%13

⎤
⎥
⎥
⎥
⎥
⎥
⎦

⎞
⎟
⎟
⎟
⎟
⎟
⎠

To get the actual inverse, you can enter:
Input:

eval(ans())

Output:

⎡
⎢
⎢
⎢
⎢
⎢
⎣

2%13

⎛ ⎝ | −4 | ⎞ ⎠ | %13

⎛ ⎝ | −5 | ⎞ ⎠ | %13

2%13

0%13

⎛ ⎝ | −5 | ⎞ ⎠ | %13

⎛ ⎝ | −2 | ⎞ ⎠ | %13

⎛ ⎝ | −1 | ⎞ ⎠ | %13

6%13

⎤
⎥
⎥
⎥
⎥
⎥
⎦

which is the inverse of A=[[1,2,9],[3,10,0],[3,11,1]] in ℤ/13ℤ.
Input (in Maple mode):

Inverse([[1,2,9],[3,10,0],[3,11,1]]) mod 13

Output:

⎡
⎢
⎢
⎣

2 | −4 | −5
2 | 0 | −5
−2 | −1 | 6

⎤
⎥
⎥
⎦