/ A11  A12 \
det |          | = A11*A22 - A12*A21
    \ A21  A22 /

    / A11  A12  A13 \
    |               |
det | A21  A22  A23 | = A11*A22*A33 + A12*A23*A31 + A13*A21*A32 - A11*A23*A32 - A13*A22*A31 - A12*A21*A33
    |               |
    \ A31  A32  A33 /

• If A, B and C are 3 square matrices such that C = A*B,  det(C) = det(A)*det(B).

• The equation a*x=b has a solution if a≠0. If a=0 it has no solution or infinitely many solutions.
Similarly, the system A*X = B has a (unique) solution if det(A) ≠ 0. If det(A) = 0 it has no solution or infinitely many solutions.

 x     - 2z =  1      / 1   0  -2 \      /  1 \      / x \
2x +  y - z = -1  A = | 2   1  -1 |  B = | -1 |  X = | y |  det(A) = 9
 x - 2y + z = -2      \ 1  -2   1 /      \ -2 /      \ z /

9 ≠ 0:  1 solution (x = -5/9, y = 1/3, z = -7/9)

 x     - 2z  =  1      / 1   0  -2 \      /  1 \      / x \
2x +  y - z  = -1  A = | 2   1  -1 |  B = | -1 |  X = | y |  det(A) = 0
 x - 2y - 8z = -2      \ 1  -2  -8 /      \ -2 /      \ z /

det(A) = 0.

 x     - 2z =  1   x = 2z+1         x = 2z+1      x = 2z+1
2x +  y - z = -1   4z+2+y-z = -1    3z+y = -3     y = -3-3z
 x - 2y -8z = -2   2z+1-2y-8z = -2  -6z-2y = -3  -6z+6+6z = -3  6 = -3   No solution

A determinant of order 5:

A determinant of order 6:

Does it exist and what is the inverse of this matrix?

/ 3 -1 0\ |-2 1 1| \ 2 -1 4/

The determinant is 5 (≠0). The inverse exists and its determinant is 1/5 = 0.2. We can find the inverse of a 3x3 matrix M by solving:
   M * (x / y / z) = (1 / 0 / 0),    M * (x / y / z) = (0 / 1 / 0),    M * (x / y / z) = (0 / 0 / 1)

The inverse is the matrix: We can find that its determinant is 0.2.

/1 0.8 -0.2\ |2 2.4 -0.6| \0 0.2 0.2/