2. Notes 7.2 – Matrices

3. I. Matrices A.) Def. – A rectangular array of numbers. An m x n matrix is a matrix consisting of m rows and n columns. The element a ij is the element found in row i, column j. B.) Matrices are only considered to be equivalent when they have the same order, and their corresponding elements are equal.

4. C.) Matrix Addition and Subtraction 1.) Let A = [a ij ] and B = [b ij ]. Then, A + B = [a ij + b ij ] and A – B = [a ij - b ij ]

5. D.) Scalar Multiplication and Properties- 1.) Let A = [a ij ]. Then, kA = k[a ij ] = [ka ij ] where k is a constant E.) Additive Identity, Zero Matrix, and Additive Inverse – See page 579

6. II. Basic Matrix Operations A.) Ex. 1 - Let A = [a ij ] and B = [b ij ] be 3 x 3 matrices with a ij = 3i – j and b ij = i 2 + j 2. 1.) Find A and B.

7. 2.) Find 3A+ 2B.

8. Then A x B = an m by n matrix where [a ij ] x [b ij ] = [c ij ] = III. Matrix Multiplication A.) Let A be an m by r matrix where A = [a ij ] and B be an r by n matrix where B = [b ij ].

9. B.) Ex. 2 - Find the product of A and B.

10. C.) Ex. 3 - Find the product of A and B. Note, a 3 x 2 cannot be multiplied by a 3 x 3 matrix. However, we can transpose matrix A to be a 2 by 3 matrix and then multiply it by B. This product is represented by A T B.

12. AI n = I n A = A - I n is the multiplicative identity IV. Identity Matrix I n A.) Def. – An n by n matrix with a main diagonal of all 1’s and zeros everywhere else.

13. B.) For real numbers, means that is the multiplicative inverse of a. A -1 is the multiplicative inverse of A.

14. V. Finding the Inverse Matrix A -1 A.) Ex. 4 – Find the multiplicative inverse of

16. There has to be an easier way to find the inverse, doesn’t there?!? The answer is YES!!!!

17. VI. Determinants and Inverses (2 by 2) A.) Def. – If ad – bc ≠ 0, then The number ad – bc is the determinant of a 2 by 2 matrix

18. B.) Ex. 5– Find the multiplicative inverse of the following matrices

19. Therefore, B has no inverse. It is called a singular matrix.

20. VII. Cofactors A.) Def. – Let A = [a ij ] be a matrix of order n x n where n > 2. M ij is the MINOR determinant. (i.e., the determinant of the (n – 1) by (n – 1) matrix obtained by deleting the row and column containing a ij.) The cofactor corresponding to a ij is

21. B.) Ex. 5– Find the all the cofactors for matrix A.

23. VII. Determinants and Inverses (n by n) A.) Thm – An n by n matrix A has an inverse iff det A ≠ 0. B.) Def. – Let A = [a ij ] be a matrix of order n x n where n > 2. The determinant of A, |A|, is the sum of the entries in any row or any column multiplied by their respective cofactors. For example, expanding by the ith row gives

24. Using the first row Using the second column B.) Ex. 6 – Determine whether matrix A has an inverse by finding the determinant. Yes, A has an inverse!

25. Rewrite the first 2 columns outside the matrix Find each diagonal product and add them C.) Another way to determine the determinant of a 3 x 3 matrix is to do the following

26. Find each of the three opposite diagonal products and add them Subtract your second product from the first and you have your determinant.

27. D.) Ex. – Find the inverse of A. Matrix – Names - x -1 - ENTER C.) The only way we have of finding the inverse of an n by n matrix for n > 3 is to use cofactors or our graphing calculator! Matrix – Edit - Quit

28. VIII. Apps: Reflection Matrices To reflect across the… multiply [x, y] by… x-axis: y-axis: origin: