Appendix II: Introduction to Matrices

Appendix II: Introduction to Matrices Find the product AB Find the the transpose of B (i.e) Def: Theorem:

Appendix II: Introduction to Matrices Find the augmented matrix Def: A matrix A in row-echelon form if • The first nonzero entry in a nonzero row is 1 • In consecutive nonzero rows the first entry 1 in the lower row appears to the right of the first 1 in the higher row • Rows consisting of all 0’s are at the bottom of the matrix

Appendix II: Introduction to Matrices Theorem: (by row operation) Row Operation • Multiply a row by a nonzero constant • Interchange any two rows • Add a nonzero constant multiple of one row to any other

Appendix II: Introduction to Matrices Def: A matrix A in reduced-row-echelon form if • A is row-echelon form • A column containing a first entry 1 has 0’s everywhere else Theorem: (by row operation)

Solving Linear System Gaussian Elimination Method: Solve: Gauss-Jordan Elimination Method: Row-echelon form Reduced Row-echelon form

Using Row operation to find the inverse Theorem: Special Case:

Minors and Cofactor to find the inverse Minors: Cofactor:

Minors and Cofactor to find the inverse Cofactor: Theorem II.2:

Minors and Cofactor to find the inverse Cofactor: Determinant: Determinant: Expand along row or column

The Eigenvalue Problem Characteristic Equation: It is a polynomial of order n. ( A is nxn) Eigenvalues of A are the roots of the characteristic equation Eigenvalues:

The Eigenvalue Problem Characteristic Equation: It is a polynomial of order n. ( A is nxn) Eigenvalues of A are the roots of the characteristic equation Eigenvalues: Eigenvalues of A are the roots of the characteristic equation Eigenvector:

Sec 2.4 Check point: function of y only How to find f(x,y) ? Given an exact DE: ----- (1) Step 1 Integrate wrt x: ----- (2) Step 2 Differentiate (2) wrt y and equate to N Step 3 ----- (3) Find: Step 3 Use (2) and (3) to write: 1 2

Appendix II: Introduction to Matrices