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, 2017 1 / 17
Question 2 1 2 Let A = 3 4 1 2 3 5 6 and B = then [BA] 1 2 3 1,3 is 7 8 A. 9 B. 11 C. 14 D. 33 E. Undefined Quang T. Bach Math 18 October 18, 2017 2 / 17
Question 3 1 2 Let A = 3 4 1 2 3 5 6 and B = then [AB] 1 2 3 1,3 is 7 8 A. 9 B. 11 C. 14 D. 33 E. Undefined Quang T. Bach Math 18 October 18, 2017 3 / 17
Question 4 A = 2 4 2 1 0 3 and B = 1 0 0 the first column of AB is 2 1 2 1 1 1 (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, 2017 4 / 17
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, 2017 5 / 17
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, 2017 5 / 17
Question 5 1 2 Let A = 3 4 then the transpose of A is 5 6 2 1 A. A T = 4 3 5 3 1 6 5 B. A T = 6 4 2 1 3 5 C. A T = 2 4 6 2 4 6 D. A T = 1 3 5 E. Can u go back one slide? Quang T. Bach Math 18 October 18, 2017 6 / 17
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, 2017 7 / 17
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, 2017 7 / 17
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, 2017 8 / 17
Finding the Inverse - 2 2 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, 2017 9 / 17
Question 6 1 2 Let A = then the inverse A 3 4 1 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 = 3/2 2 2 1 D. A 1 = 3/2 1/2 E. The matrix is not invertible Quang T. Bach Math 18 October 18, 2017 10 / 17
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, 2017 11 / 17
An Algorithm for Finding A 1 - Examples Example 1 1 0 Find the inverse of the matrix A = 1 0 1 6 2 3 Quang T. Bach Math 18 October 18, 2017 12 / 17
An Algorithm for Finding A 1 - Examples Example 1 1 0 Find the inverse of the matrix A = 1 0 1 6 2 3 Step 1: Form the augmented matrix 1 1 0. 1 0 0 A I = 1 0 1. 0 1 0 6 2 3. 0 0 1 Quang T. Bach Math 18 October 18, 2017 12 / 17
An Algorithm for Finding A 1 - Examples Step 2: Row reduce [ A I ] 1 1 0 1 0 0 1 0 0 2 3 1 A I = 1 0 1 0 1 0 0 1 0 3 3 1 6 2 3 0 0 1 0 0 1 2 4 1 Quang T. Bach Math 18 October 18, 2017 13 / 17
An Algorithm for Finding A 1 - Examples Step 3: Decide if A is invertible and find the inverse 1 0 0. 2 3 1 A I 0 1 0. 3 3 1 = [ I A 1] 0 0 1. 2 4 1 2 3 1 So A is invertible with A 1 = 3 3 1 2 4 1 Quang T. Bach Math 18 October 18, 2017 14 / 17
An Algorithm for Finding A 1 - Examples Let s try another example Example 1 2 0 Find the inverse of the matrix A = 3 1 2 2 3 2 Quang T. Bach Math 18 October 18, 2017 15 / 17
An Algorithm for Finding A 1 - Examples Let s try another example Example 1 2 0 Find the inverse of the matrix A = 3 1 2 2 3 2 Step 1: Form the augmented matrix 1 2 0 A I = 3 1 2. 1 0 0. 0 1 0 2 3 2. 0 0 1 Quang T. Bach Math 18 October 18, 2017 15 / 17
An Algorithm for Finding A 1 - Examples Step 2: Row reduce [ A I ] 1 2 0 1 0 0 1 2 0 1 0 0 A I = 3 1 2 0 1 0 0 7 2 3 1 0 2 3 2 0 0 1 0 7 2 2 0 1 1 2 0 1 0 0 0 7 2 3 1 0 0 0 0 1 1 1 Quang T. Bach Math 18 October 18, 2017 16 / 17
An Algorithm for Finding A 1 - Examples Step 3: Decide if A is invertible and find the inverse 1 2 0 1 0 0 A I 0 7 2 3 1 0 0 0 0 1 1 1 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, 2017 17 / 17