If A is a 4 6 matrix and B is a 6 3 matrix then the dimension of AB is A. 4 6 B. 6 6 C. 4 3 D. 3 4 E. Undefined

Transcription

1 Question 1 If A is a 4 6 matrix and B is a 6 3 matrix then the dimension of AB is A. 4 6 B. 6 6 C. 4 3 D. 3 4 E. Undefined Quang T. Bach Math 18 October 18, / 17

2 Question Let A = and B = then [BA] ,3 is 7 8 A. 9 B. 11 C. 14 D. 33 E. Undefined Quang T. Bach Math 18 October 18, / 17

3 Question Let A = and B = then [AB] ,3 is 7 8 A. 9 B. 11 C. 14 D. 33 E. Undefined Quang T. Bach Math 18 October 18, / 17

4 Question 4 A = and B = the first column of AB is (A.) 5 3 (B.) 5 6 (C.) 5 8 (E.) Idk I didn t do what u asked (D.) Undefined Quang T. Bach Math 18 October 18, / 17

5 Transpose of a Matrix Definition Let A be an m n matrix. The transpose of a matrix A, denote A T is an n m whose columns are formed from the corresponding rows of A. Quang T. Bach Math 18 October 18, / 17

6 Transpose of a Matrix Definition Let A be an m n matrix. The transpose of a matrix A, denote A T is an n m whose columns are formed from the corresponding rows of A. Theorem Let A and B be matrices with dimensions such that the following matrix sums and products are defined, and let c be any scalar. (A T ) T = A (A + B) T = A T + B T (ca) T = c(a T ) (AB) T = B T A T (notice the reverse order!) Quang T. Bach Math 18 October 18, / 17

7 Question Let A = 3 4 then the transpose of A is A. A T = B. A T = C. A T = D. A T = E. Can u go back one slide? Quang T. Bach Math 18 October 18, / 17

8 Inverse of a Matrix - Definition Definition A square n n matrix is called invertible (or non-singular) if there exists an n n matrix B such that AB = BA = I n where I n is the identity matrix of order n. The matrix B is called the (multiplicative) inverse of A. If there is no such matrix B exists then A does not have an inverse, and is called non-invertible (or singular). Quang T. Bach Math 18 October 18, / 17

9 Inverse of a Matrix - Definition Definition A square n n matrix is called invertible (or non-singular) if there exists an n n matrix B such that AB = BA = I n where I n is the identity matrix of order n. The matrix B is called the (multiplicative) inverse of A. If there is no such matrix B exists then A does not have an inverse, and is called non-invertible (or singular). Remarks: Invertible only applies to square matrices We can show that of A is invertible and the inverse is unique. Thus, we can write the (unique) inverse of A as A 1 Quang T. Bach Math 18 October 18, / 17

10 Inverse of a Matrix - Properties Theorem a. If A is invertible then A 1 is also invertible and (A 1 ) 1 = A b. If A, B are n n invertible matrix then so is AB. The inverse of AB is given by (AB) 1 = B 1 A 1 (again, notice the reverse order) c. If A is an invertible matrix then so is A T. The inverse of A T is given by (A T ) 1 = (A 1 ) T Quang T. Bach Math 18 October 18, / 17

11 Finding the Inverse Case Theorem a b Let A = be a 2 2 matrix. Then c d A is invertible ad bc 0 When A is invertible, the inverse is given by A 1 1 d b = ad bc c a Quang T. Bach Math 18 October 18, / 17

12 Question Let A = then the inverse A is 1/2 1 A. A 1 = 3/2 2 1/2 1 B. A 1 = 3/2 2 1/2 1 C. A 1 = 3/ D. A 1 = 3/2 1/2 E. The matrix is not invertible Quang T. Bach Math 18 October 18, / 17

13 An Algorithm for Finding A 1 1 Form the augmented matrix [ A I ] 2 Use Gauss-Jordan Elimination to find the reduced row echelon form (rref) of [ A I ] 3 Observe the left part of the rref. i. If we get the identity matrix I on the left of the rref, then [ A I ] [ I A 1 ]. That is, we can obtain A 1 by reading the right of the rref. ii. Otherwise, A is not invertible. Quang T. Bach Math 18 October 18, / 17

14 An Algorithm for Finding A 1 - Examples Example Find the inverse of the matrix A = Quang T. Bach Math 18 October 18, / 17

15 An Algorithm for Finding A 1 - Examples Example Find the inverse of the matrix A = Step 1: Form the augmented matrix A I = Quang T. Bach Math 18 October 18, / 17

16 An Algorithm for Finding A 1 - Examples Step 2: Row reduce [ A I ] A I = Quang T. Bach Math 18 October 18, / 17

17 An Algorithm for Finding A 1 - Examples Step 3: Decide if A is invertible and find the inverse A I = [ I A 1] So A is invertible with A 1 = Quang T. Bach Math 18 October 18, / 17

18 An Algorithm for Finding A 1 - Examples Let s try another example Example Find the inverse of the matrix A = Quang T. Bach Math 18 October 18, / 17

19 An Algorithm for Finding A 1 - Examples Let s try another example Example Find the inverse of the matrix A = Step 1: Form the augmented matrix A I = Quang T. Bach Math 18 October 18, / 17

20 An Algorithm for Finding A 1 - Examples Step 2: Row reduce [ A I ] A I = Quang T. Bach Math 18 October 18, / 17

21 An Algorithm for Finding A 1 - Examples Step 3: Decide if A is invertible and find the inverse A I Because the left part has a row of zeros, we can conclude that it is not possible to reduce [ A I ] into [ I A 1]. So A is not invertible. Quang T. Bach Math 18 October 18, / 17