A determinant is a square array of numbers.

| a11 a21 a31 a41 |
|                 |
| a21 a22 a32 a42 |
|                 |
| a31 a23 a33 a43 |
|                 |
| a41 a24 a34 a44 |

Each number is an element of the determinant. The determinant is classified to the square.
A determinant square(4) is a fourth-order determinant. The determinant has a single value.

The basic determinant is a second-order determinant.

| a1,1  a1,2 |
|            |
| a2,1  a2,2 |

All determinants can be expressed as terms of second-order determinants.

The value D, of a second-order determinant is defined as:

D = ((a1,1)*(a2,2))-((a1,2)*(a2,1))

As a multidimensional array:

D = a[1][1]*a[2][2]-a[1][2]*a[2][1] 

So:

| 3  5 |
|      |
| 6  2 |

is:

D = (3*2) - (5*6) = -24

Every element of a higher-order determinant (Nth-order determinant) has a MINOR determinant. To find the minor determinant of an element of a higher-order determinant, strike out the row and column of the element of the higher-order determinant.

Thus the determinant of the element a[1][2]:

| 3  5  2 |
|         |
| 6  2  6 |
|         |
| 1  8  7 |

is the determinant:

| -  -  - |
|         |
| 6  -  6 |
|         |
| 1  -  7 |

So of element a[1][2], the value of the minor determinant is:

D = (6*7) - (6*1) = 42 - 6 = 36

This can be done for any element a[i][j]

( row i , column j )

M[i][j] = minor determinant of a[i][j]

With the MINOR M[i][j] we can denote the cofactor A[i][j] of element a[i][j].

 cofactor a[i][j] = A[i][j]
A[i][j] = (-1)^i+j * M[i][j]

Thus the cofactor of a[1][1] is:

i = 1
j = 1
i + j = 2
-1^2 = -1 * -1 = 1
M[i][j] = (2*7)-(6*8) = 14-48 = -34
A[i][j] = 1 * -34 = -34

Thus the cofactor of a[1][2] is:

i = 1
j = 2
i + j = 3
-1^3 = -1 * -1 * -1 = -1
M[i][j] = (6*7)-(6*1) = 42-6 = 36
A[i][j] = -1 * 36 = -36

Thus the cofactor of a[1][3] is:

i = 1
j = 3
i + j = 4
-1^3 = -1 * -1 * -1 * -1 = 1
M[i][j] = (6*8)-(2*1) = 48-2 = 46
A[i][j] = 1 * 46 = 46

Now:

D = 3*-34 + 5*-36 + 2*46
D = -102 + -180 + 92
D = -190