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