1. 1 Matrix Math ©Anthony Steed 1999

2. 2 Overview n To revise Vectors Matrices

3. 3 Vectors and Matrices n Matrix is an array of numbers with dimensions M (rows) by N (columns) 3 by 6 matrix element 2,3 is (3) n Vector can be considered a 1 x M matrix

4. 4 Row 1 Row 2 Row 3 Row m Column 1 Column 2 Column 3 Column 4

5. 5 A matrix of m rows and n columns is called a matrix with dimensions m x n. 2 X 3 3 X 3 2 X 1 1 X 2

6. 6 Types of Matrix n Identity matrices - I n Diagonal n Symmetric Diagonal matrices are (of course) symmetric Identity matrices are (of course) diagonal

7. 7 Operation on Matrices n Addition Done elementwise n Transpose “Flip” (M by N becomes N by M)

8. 8 Operations on Matrices n Multiplication Only possible to multiply of dimensions –x 1 by y 1 and x 2 by y 2 iff y 1 = x 2 resulting matrix is x 1 by y 2 –e.g. Matrix A is 2 by 3 and Matrix by 3 by 4 resulting matrix is 2 by 4 –Just because A x B is possible doesn’t mean B x A is possible!

9. 9 Matrix Multiplication Order n A is n by k, B is k by m n C = A x B defined by n BxA not necessarily equal to AxB

10. 10 Example Multiplications

11. 11 Inverse n If A x B = I and B x A = I then A = B -1 and B = A -1

12. 12 Scale n Scale uses a diagonal matrix n Scale by 2 along x and -2 along z

13. 13 DETERMINANT A determinant is a value that is obtained from a square matrix.

14. 14 n SARRUS RULE

15. 15 CRAMER RULE

16. 16 Using Determinants to Solve Systems of Equations. – Systems of Linear Equations - Determinants

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