Lesson 54 – Multiplication of Matrices Math 2 Honors - Santowski IB Math SL - Santowski

Lesson Objectives (1) Review simple terminology associated with matrices (2) Review simple operations with matrices (+,-, scalar multiplication) (3) Compare properties of numbers with matrices (and at the same time introduce the use of the GDC) (4) Multiply matrices IB Math SL - Santowski

(F) Properties of Matrix Addition Ordinary Algebra with Real numbers Matrix Algebra if a and b are real numbers, so is the product a+b Commutative: a + b = b + a for all a,b Associative: (a + b) + c = a + (b + c) Additive Identity: a + 0 = 0 + a = a for all a Additive Inverse: a + (-a) = (-a) + a = 0 IB Math SL - Santowski

TI-84 and Matrices Here are the screen captures on HOW to use the TI-84 wherein we test our properties of matrix addition Use 2nd x-1 to access the matrix menu IB Math SL - Santowski

(F) Properties of Matrix Addition Ordinary Algebra with Real numbers Matrix Algebra if a and b are real numbers, so is the product a+b if A and B are matrices, so is the sum A + B provided that ….. Commutative: a + b = b + a for all a,b in general, A + B = B + A provided that ….. Associative: (a + b) + c = a + (b + c) (A + B) + C = A + (B + C) is true provided that ….. Additive Identity: a + 0 = 0 + a = a for all a A + 0 = 0 + A = A for all A where 0 is the zero matrix Additive Inverse: a + (-a) = (-a) + a = 0 A + (-A) = (-A) + A = 0 IB Math SL - Santowski

Multiplying Matrices - Generalized Example If we multiply a 2×3 matrix with a 3×1 matrix, the product matrix is 2×1 Here is how we get M11 and M22 in the product. M11 = r11× t11  +  r12× t21  +   r13×t31 M12 = r21× t11  +  r22× t21   +  r23×t31 IB Math SL - Santowski

(B) Matrix Multiplication - Summary Summary of Multiplication process IB Math SL - Santowski

(D) Examples for Practice Multiply the following matrices: IB Math SL - Santowski

(D) Examples for Practice Multiply the following matrices: IB Math SL - Santowski

(E) Examples for Practice – TI-84 Here are the key steps involved in using the TI-84 IB Math SL - Santowski

(E) Examples for Practice – TI-84 Here are the key steps involved in using the TI-84 IB Math SL - Santowski

(F) Properties of Matrix Multiplication Now we pass from the concrete to the abstract  What properties are true of matrix multiplication where we simply have a matrix (wherein we know or don’t know what elements are within) Asked in an alternative sense  what are the general properties of multiplication (say of real numbers) in the first place??? IB Math SL - Santowski

(F) Properties of Matrix Multiplication Ordinary Algebra with Real numbers Matrix Algebra if a and b are real numbers, so is the product ab ab = ba for all a,b a0 = 0a = 0 for all a a(b + c) = ab + ac a x 1 = 1 x a = a an exists for all a > 0 IB Math SL - Santowski

(C) Key Terms for Matrices We learned in the last lesson that there is a matrix version of the addition property of zero. There is also a matrix version of the multiplication property of one. The real number version tells us that if a is a real number, then a*1 = 1*a = a. The matrix version of this property states that if A is a square matrix, then A*I = I*A = A, where I is the identity matrix of the same dimensions as A. Definition  An identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere. IB Math SL - Santowski

(C) Key Terms for Matrices Definition  An identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere. So, in matrix multiplication  A x I = I x A = A IB Math SL - Santowski

(F) Properties of Matrix Multiplication Ordinary Algebra with Real numbers Matrix Algebra if a and b are real numbers, so is the product ab if A and B are matrices, so is the product AB ab = ba for all a,b in general, AB ≠ BA a0 = 0a = 0 for all a A0 = 0A = 0 for all A where 0 is the zero matrix a(b + c) = ab + ac A(B + C) = AB + AC a x 1 = 1 x a = a AI = IA = A where I is called an identity matrix and A is a square matrix an exists for all a > 0 An for {n E I | n > 2} and A is a square matrix IB Math SL - Santowski

(F) Properties of Matrix Multiplication This is a good place to use your calculator if it handles matrices. Do enough examples of each to convince yourself of your answer to each question (1) Does AB = BA for all B for which matrix multiplication is defined if ? (2) In general, does AB = BA? (3) Does A(BC) = (AB)C? (4) Does A(B + C) = AB + AC? IB Math SL - Santowski

(F) Properties of Matrix Multiplication This is a good place to use your calculator if it handles matrices. Do enough examples of each to convince yourself of your answer to each question (6) Does A - B = -(B - A)? (7) For real numbers, if ab = 0, we know that either a or b must be zero. Is it true that AB = 0 implies that A or B is a zero matrix? IB Math SL - Santowski

Internet Links http://www.intmath.com/matrices-determinants/3-matrices.php http://www.purplemath.com/modules/mtrxadd.htm IB Math SL - Santowski