### 11.9.5 Determinant of a matrix with coefficients in ℤ/pℤ

In Xcas mode, Det is simply the inert form of det; namely, it gives the determinant of a matrix without evaluating it. (See Section 15.1.4.) In Maple mode, the Det command can additionally be used in conjunction with mod to find the determinant of a matrix whose elements are in ℤ/pℤ.

- In Maple mode, Det takes A, a matrix with elements in ℤ/pℤ.
- Det(A) returns the determinant of A.

#### Example

Input in Xcas mode:

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

det | ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ |  |
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
1%13 | 2%13 |
⎛ ⎝ | −4 | ⎞ ⎠ | %13
3%13 |
⎛ ⎝ | −3 | ⎞ ⎠ | %13
| 0%13
3%13 |
⎛ ⎝ | −2 | ⎞ ⎠ | %13
| 1%13
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
|  | ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

To find the value of the determinant, enter:

eval(ans())

5%13 |  |  |  |  |  |  |  |  |  |

Hence, in ℤ/13ℤ, the determinant of A=[[1, 2, 9],[3,10,0],[3,11,1]] is 5%13 (in ℤ, det(A)=31).

Input in Maple mode:

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

5 |  |  |  |  |  |  |  |  |  |