Matrices

Matrices

1. Find determinant

   • Co-factor expansion
   • Diagonals
     • 2x2
     • 3x3
   • Facts to make det easier
     • Det of upper or lower triangular matrix is product of entries on main diagonal
       • Operations to get to triangular form
     • det(A)=det(A transpose)
     • If A has a row/column of 0's then det(A)=0
     • det(AB)=det(A)det(B)

2. Applications

   • Eigenvectors and eigenvalues
     • Eigenvector cannot be 0 vector
   • Solving Ax=b for x
     • Cramer's rule
     • Gaussian elimination on augmented matrix to upper triangular form then use back substitution
     • Find inverse A and times by b

3. Algebra

   • Transpose swaps rows and columns
   • Add/subtract - allowed if same size
   • Multiplication e.g. AB - allowed if column no of A = row no of B
     • Matrix multiplication is not commutative
   • AA^-1 = I For square only
   • Singular = non-invertible

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