Determinants - Examples Involving Determinats And Inverse Of Matrices

Determinants - Examples involving determinants and inverse of Matrices

Recap of determinants and inverse of matrices
Determinant of a matrix
- Definition and notation
- Properties of determinants
  1. Scalar multiplication
  2. Row operation
  3. Column operation
  4. Interchange of rows or columns
  5. Determinant of an identity matrix
  6. Determinant of a transpose matrix
  7. Determinant of a product of matrices
Inverse of a matrix
- Definition and notation
- Conditions for existence of inverse
- Method to find inverse using adjoint and determinant
Example: Finding determinant and inverse of a 3x3 matrix

Example 1: Find the determinant of the matrix A, given by:
- A = [1 2; 3 4]
Example 2: Find the inverse of the matrix B, given by:
- B = [5 6; 7 8]
Example 3: Find the determinant and inverse of the matrix C, given by:
- C = [2 -1 3; 4 2 -1; 1 3 2]
Example 4: Find the determinant and inverse of the matrix D, given by:
- D = [1 2 3; 4 5 6; 7 8 9]

Example 5: Find the determinant and inverse of the matrix E, given by:
- E = [2 0 1; 3 1 2; 1 4 3]
Example 6: Find the determinant and inverse of the matrix F, given by:
- F = [1 3; 2 5]
Example 7: Find the determinant and inverse of the matrix G, given by:
- G = [3 -1 2; 4 1 -3; 2 -2 1]
Example 8: Find the determinant and inverse of the matrix H, given by:
- H = [-1 2 -3; 4 -5 6; -7 8 -9]
Example 9: Find the determinant and inverse of the matrix I, given by:
- I = [2 3 1; 5 1 -2; 3 2 4]
Example 10: Find the determinant and inverse of the matrix J, given by:
- J = [1 -2 3; -4 5 -6; 7 -8 9]

Example 11: Find the determinant and inverse of the matrix K, given by:
- K = [-1 -2; 3 4]
Example 12: Find the determinant and inverse of the matrix L, given by:
- L = [2 1 -3; 1 3 -2; -3 -2 1]
Example 13: Find the determinant and inverse of the matrix M, given by:
- M = [2 -1; -3 4]
Example 14: Find the determinant and inverse of the matrix N, given by:
- N = [1 2 3; 0 1 4; -1 0 1]
Example 15: Find the determinant and inverse of the matrix O, given by:
- O = [3 4; 2 1]

Example 16: Find the determinant and inverse of the matrix P, given by:
- P = [2 -3 1; 3 2 -1; 1 1 1]
Example 17: Find the determinant and inverse of the matrix Q, given by:
- Q = [-2 0 3; 1 -1 2; 3 4 -5]
Example 18: Find the determinant and inverse of the matrix R, given by:
- R = [1 1 -2 -3; 3 4 1 2; -1 -2 2 3; 1 -4 -2 1]
Example 19: Find the determinant and inverse of the matrix S, given by:
- S = [1 2 3 4; 0 1 2 3; 1 0 1 2; 1 1 0 1]
Example 20: Find the determinant and inverse of the matrix T, given by:
- T = [2 1 4; -3 -1 -2; 1 -4 3]

Example 21:
- Given matrix U:
  - U = [1 -2 3; 0 1 4; -1 0 1]
- Find the determinant and inverse of U.
Example 22:
- Given matrix V:
  - V = [3 1 2; 0 2 3; 1 -2 1]
- Find the determinant and inverse of V.
Example 23:
- Given matrix W:
  - W = [2 -1 3 0; 1 -3 2 -1; -3 -2 1 2; 3 4 -5 6]
- Find the determinant and inverse of W.
Example 24:
- Given matrix X:
  - X = [4 0 1 0; -1 3 -2 1; 2 -3 4 -2; 0 2 -1 1]
- Find the determinant and inverse of X.
Example 25:
- Given matrix Y:
  - Y = [1 2 3 4 5; 0 1 2 3 4; 1 0 1 2 3; 1 1 0 1 2; 1 1 1 1 1]
- Find the determinant and inverse of Y.

Example 26:
- Given matrix Z:
  - Z = [2 -3 1 4; 3 2 -1 2; 1 1 1 3; 4 2 3 -1]
- Find the determinant and inverse of Z.
Example 27:
- Given matrix A1B1C1D1:
  - A1B1C1D1 = [1 1 1 1; 1 2 3 4; 1 3 6 10; 1 4 10 20]
- Find the determinant and inverse of A1B1C1D1.
Example 28:
- Given matrix E1F1G1H1:
  - E1F1G1H1 = [2 -1 3 2; -1 2 -3 1; 3 -3 2 -2; 2 1 -2 1]
- Find the determinant and inverse of E1F1G1H1.
Example 29:
- Given matrix I1J1K1L1:
  - I1J1K1L1 = [1 2 3 4; 2 1 4 3; 3 4 1 2; 4 3 2 1]
- Find the determinant and inverse of I1J1K1L1.
Example 30:
- Given matrix M1N1O1P1:
  - M1N1O1P1 = [1 4 7 10; 4 2 8 1; 7 8 3 4; 10 1 4 9]
- Find the determinant and inverse of M1N1O1P1.