Lesson 55 Inverse of Matrices Determinants Math 2

Lesson 55 - Inverse of Matrices & Determinants Math 2 Honors - Santowski 1

(A) Review n - at this stage of studying matrices, we know how to add, subtract and multiply matrices n i. e. if n Then evaluate: (a) A + B (b) -3 A (c) BA (d) B – A (e) AB n n n Math 2 Honors - Santowski 2

(B) Review of Real Numbers n if we divide 5 by 8 (i. e. 5/8), we could rearrange and look at division as nothing more than simple multiplication n thus 5/8 = 5 x 1/8 = 5 x 8 -1 n so in a way, we would never have to perform division as long as we simply multiply by the inverse (or reciprocal) n n One other note about this inverse of a number and its inverse (its reciprocal) have the property that (n) x (n-1) = 1 - i. e. (8) (8 -1) = (8) (1/8) = (8/8) = 1 n So how does this relate to DIVISION of MATRICES? ? Math 2 Honors - Santowski 3

(C) Strategy for “Dividing” Matrices n So how does “multiplicative inverses” relate to DIVISION of MATRICES? ? n If a number and its inverse (its reciprocal) have the property that (n) x (n-1) = 1 n Then …. Math 2 Honors - Santowski 4

(C) Strategy for “Dividing” Matrices n So how does “multiplicative inverses” relate to DIVISION of MATRICES? ? n If a number and its inverse (its reciprocal) have the property that (n) x (n-1) = 1 n Then …. a matrix and its “inverse” should have the property that B x B -1 = 1 Math 2 Honors - Santowski 5

(C) Strategy for “Dividing” Matrices n So …. a matrix and its “inverse” should have the property that B x B -1 = 1 n Well what is 1 in terms of matrices? simply the identity matrix, I n Thus B x B -1 = I Math 2 Honors - Santowski 6

(D) Inverse Matrices n Given matrix A, which of the following 4 is the inverse of matrix A? Math 2 Honors - Santowski 7

(D) Inverse Matrices n Solve for x: Math 2 Honors - Santowski 8

(E) Terms Associated with Inverse Matrices n Thus we have 2 new terms that relate to inverse matrices: n (a) a matrix is invertible if it has an inverse n (b) a matrix is singular if it does NOT have an inverse Math 2 Honors - Santowski 9

(F) Inverse Matrices on TI-83/4 n n So we have the basic idea of inverse matrices how can I use the calculator to find the inverse of a matrix? ? Math 2 Honors - Santowski 10

(F) Inverse Matrices on TI-83/4 n Use the TI-83/4 to determine the inverse of: Math 2 Honors - Santowski 11

(G) Properties of Inverses (and Matrix Multiplication) n Is multiplication with real numbers commutative (is ab = ba)? n Is matrix multiplication commutative q n Is AB = BA? (use TI-84 to investigate) Is A x A-1 = A-1 x A = I? (use TI-84 to investigate) Math 2 Honors - Santowski 12

(G) Properties of Inverses (and Matrix Multiplication) n Are these “properties” true for (i) real numbers? (ii) matrices? ? ? Use TI-84 to investigate n Is (A-1)-1= A ? ? ? n Is (AB)-1 = A-1 B-1 ? Math 2 Honors - Santowski 13

(H) Determining the Inverse of a Matrix n How can we determine the inverse of a matrix if we DO NOT have access to our calculators? n (i) Matrix Multiplication (ii) Calculating the “determinant” n Math 2 Honors - Santowski 14

(H) Determining the Inverse of a Matrix n Let’s use Matrix Multiplication to find the inverse of n So our matrix will be n And we now have the multiplication n And so using our knowledge of matrix multiplication, we get Math 2 Honors - Santowski 15

(H) Determining the Inverse of a Matrix n And so using our knowledge of matrix multiplication, we get a system of 4 equations n Which we can solve as: Math 2 Honors - Santowski 16

(H) Determining the Inverse of a Matrix n So if n So our matrix will be Math 2 Honors - Santowski 17

(H) Determining the Inverse of a Matrix n How can we determine the inverse of a matrix if we DO NOT have access to our calculators? n (ii) Calculating the “determinant” n So Method #2 involved something called a “determinant” which means …. . ? ? Math 2 Honors - Santowski 18

(I) Determinants An Investigation n Use your TI-83/4 to determine the following products: Math 2 Honors - Santowski 19

(I) Determinants An Investigation n Use your TI-83/4 to determine the following products: Math 2 Honors - Santowski 20

(I) Determinants An Investigation n Now carefully look at the 2 matrices you multiplied and observe a pattern ? ? Math 2 Honors - Santowski 21

(I) Determinants An Investigation n Now carefully look at the 2 matrices you multiplied and observe a pattern ? ? Math 2 Honors - Santowski 22

(I) Determinants An Investigation n Now PROVE your pattern holds true for all values of a, b, c, d …. Math 2 Honors - Santowski 23

(I) Determinants An Investigation n n Now PROVE your pattern holds true for all values of a, b, c, d …. Math 2 Honors - Santowski 24

(I) Determinants An Investigation n So to summarize: Math 2 Honors - Santowski 25

(I) Determinants An Investigation n then we see that from our original matrix, the value (ad-bc) has special significance, in that its value determines whether or not matrix A can be inverted n -if ad - bc does not equal 0, matrix A would be called "invertible“ n - i. e. if ad - bc = 0, then matrix A cannot be inverted and we call it a singular matrix n - the value ad - bc has a special name it will be called the determinant of matrix A and has the notation det. A or |A| Math 2 Honors - Santowski 26

(I) Determinants An Investigation n So if A is invertible then Math 2 Honors - Santowski 27

(J) Examples n n ex 1. Find the determinant of the following matrices and hence find their inverses: n Verify using TI-83/4 Math 2 Honors - Santowski 28

(J) Examples n n ex 2. Find the determinant of the following matrices and hence find their inverses: n Verify using TI-83/4 Math 2 Honors - Santowski 29

(L) Homework n HW – n S 4. 3; p 239; Q 5, 6, 8, 9, 13 -35 odd, 45 Math 2 Honors - Santowski 30

(J) Examples n Prove whether the following statements are true or false for 2 by 2 matrices. Remember that a counterexample establishes that a statement is false. n In general, you may NOT assume that a statement is true for all matrices because it is true for 2 by 2 matrices, but for the examples in this question, those that are true for 2 by 2 matrices are true for all matrices if the dimensions allow the operations to be performed. n Questions: Math 2 Honors - Santowski 31