Algebra, Week 2

Matrix Inversion and Determinants

1. Matrix A and Its Adjoint

• A = (: a 2)

• The adjoint of matrix A (denoted A* or Adj A) is given as:

  • For 2x2 matrix A = ( \begin{pmatrix} a & b \ c & d \end{pmatrix} )

  • A* = ( \begin{pmatrix} d & -b \ -c & a \end{pmatrix} )

2. Determinant of Matrix A

• The determinant denoted as |A| of a 2x2 matrix A is calculated as:

  • |A| = ad - bc

3. Example Calculations

Calculating A*, AA, and det(A):

• Let ( A = \begin{pmatrix} 2 & 4 \ -3 & 1 \end{pmatrix} )

• Compute A*:

  • A* = ( \begin{pmatrix} 1 & -4 \ 3 & 2 \end{pmatrix} )

• Compute AA:

  • AA = A \times A* = ( \begin{pmatrix} 2 & 4 \ -3 & 1 \end{pmatrix} \begin{pmatrix} 1 & -4 \ 3 & 2 \end{pmatrix} )

  • Calculate elements step-by-step:

    • (1,1): 21 + 43 = 14

    • (1,2): 2*(-4) + 4*2 = 0

    • (2,1): -31 + 13 = 0

    • (2,2): -3*(-4) + 1*2 = 14

4. Properties of Matrix Products

• If AB = I (identity matrix), then B is the inverse of A (A^').

• It is known that A A* = det(A) I

• Also, A* A = det(A) I.

5. Determinant Properties

• For 2x2 matrices A and B:

  • det(AB) = det(A) * det(B)

  • If det(A) ≠ 0, then A is invertible.

  • det(A^*) = (det(A))^n, where n is the dimension of the matrix.

6. Further Examples

• Let A = ( \begin{pmatrix} 10 & -11 \ -2 & 2 \end{pmatrix} )

  • Calculate det(A): det A = 10*2 - (-11)(-2) = 20 - 22 = -2

  • Find adjoint A* and verify properties.

7. Calculating Determinants of Scaled Matrices

• For a matrix A, det(kA) = k^n * det(A) for a scalar k and an n x n matrix.

8. Linear Transformations

• Linear Transformation (LT) defined as:

  • T: R^n -> R^m such that T(u + v) = T(u) + T(v) and T(du) = dT(u) where d is a scalar.

• Can be represented using a matrix A where T(v) = A * v.

Conclusion

• Understanding the properties of matrices, their determinants, and adjoints is crucial in linear algebra for solving systems of equations and transformations.