T 4 3 Inverse of Matrices Determinants IB

T 4. 3 - Inverse of Matrices & Determinants IB Math SL - Santowski

(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

(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? ?

(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 ….

(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

(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

(D) Inverse Matrices n Given matrix A, which of the following 4 is the inverse of matrix A?

(D) Inverse Matrices n Solve for x:

(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

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

(F) Inverse Matrices on TI-83/4 n Use the TI-83/4 to determine the inverse of:

(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)

(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 ?

(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

(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

(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:

(H) Determining the Inverse of a Matrix n So if n So our matrix will be n Block D end

(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 …. . ? ?

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

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

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

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

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

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

(I) Determinants An Investigation n So to summarize:

(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|

(I) Determinants An Investigation n So if A is invertible then

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

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

(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:

(L) Homework n HW – n n Ex 14 H #2 ad, 8 acf; Ex 14 I #1 a, 3 ab, 4 b, 7; Ex 14 K #2 a; Ex 14 L #5 a, 8; n IB Packet #2, 7 n n

3 x 3 Matrices & Determinants n So far, we have worked with 2 x 2 matrices to explain/derive the concept of inverses and determinants n But what about 3 x 3 matrices? ? n Do they have inverses? How do I find the inverse? How do I calculate the determinant?

3 x 3 Matrices & Determinants n If A is the 2 by 2 matrix , then det(A) = ad - bc is found this way: n So the product of one diagonal (ad) minus the product of another diagonal (bc)

3 x 3 Matrices & Determinants n This “diagonal” trick can also be applied to 3 x 3 matrices n We will NOT attempt to PROVE it in any way in this course though you should simply be aware of a non-calculator method for finding a determinant of a 3 x 3 matrix

3 x 3 Matrices & Determinants n Let n And let’s use this diagonal difference idea …. n 4*2*6 – 1*2*3 = 48 – 6 = 42 n … but I haven’t used all the elements of the matrix …. . So …. n

3 x 3 Matrices & Determinants n Let n And let’s use this diagonal difference idea …. n 2*4*1 – 3*4*4 = 8 – 48 = -40 n And ….

3 x 3 Matrices & Determinants n Let n And let’s use this diagonal difference idea …. n 3*0*3 – 6*0*2 = 0 … so … n (42) + (-40) + (0) = 2 n

3 x 3 Matrices & Determinants n n Let n And let’s use this diagonal difference idea …. and (42) + (-40) + (0) = 2 So det[A] = 2 n n And verifying on the TI 83/4 …

3 x 3 Matrices & Determinants n There is an alternative approach to finding the determinant of a 3 x 3 matrix n The formula is n if

3 x 3 Matrices & Determinants n So working the formula:

3 x 3 Matrices & Determinants n So, if we can find a value for the determinant, what does that mean n It simply means that our original matrix is invertible and as long as det[A] ≠ 0, then we can invert our matrix and make use of the inverse

Practice n Find the determinants of these matrices. Show your work.

Homework n HW n n Ex 14 H #5; Ex 14 I #5 ac, 6, 8 b; Ex 14 J #1 agh, 3, 6 a; n IB Packet #1, 4, 5 n