Review Let A, B, and C be matrices of the same size, and let r and s be scalars. Then
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1 1 Sec 21 Matrix Operations Review Let A, B, and C be matrices of the same size, and let r and s be scalars Then (i) A + B = B + A (iv) r(a + B) = ra + rb (ii) (A + B) + C = A + (B + C) (v) (r + s)a = ra + sa (iii) A + O = A = O + A (vi) r(sa) = (rs)a Here O is the zero matrix having the same size as A Let P, Q, and R be matrices for which the indicated sums and products are well-defined in the following Let r be a scalar Then (vii) P (QR) = (P Q)R (ix) P (Q + R) = P Q + P R (viii) (P + Q)R = P R + QR (x) r(p Q) = (rp )Q = P (rq) Definition An n n matrix is called an n-dimensional square matrix The n-dimensional square matrix with 1 s on the diagonal and 0 s elsewhere is called the n-dimensional identity matrix and denoted by I n Ex is the 3-dimensional identity matrix Ex2 Compute the following: = [ = In general, for any m n matrix A, I m A = A = AI n Problem Suppose A is an n n square matrix and there exist n n square matrices C and D such that CA = I n = AD Prove that C = D Definition If A is an n n matrix and if k is a positive integer, then A k denotes the product of k copies of A: A k = AA }{{ A} k copies
2 2 2 1 Ex3 For A =, we have A 2 = = A 3 = A 2 A = Definition Given an m n matrix A, the transpose of A is the n m matrix, denoted by A T, whose columns are formed from the corresponding rows of A Ex4 A = = A T =, and B = [ = B T = Remark When A = A T (This implies A must be a square matrix), A is called a symmetric matrix Ex5 and are symmetric Properties Let A and B have sizes for which the indicated sum and product are defined, then (i) (A T ) T = A (ii) (A + B) T = A T + B T (iii) For any scalar r, (ra) T = ra T (iv) (AB) T = B T A T [ Ex6 Let A = and B = AB =, so (AB) T =, then On the other hand, A T = and B T = and hence B T A T =
3 3 Sec 22 The Inverse of a Matrix Definition An n n matrix A is said to be invertible (or nonsingular) if there is an n n matrix C such that CA = I n = AC In this case, C is called an inverse of A Remark Once A is invertible, its inverse is uniquely determined: if B were another inverse of A, then B = BI n = B(AC) = (BA)C = I n C = C From now on, when A is invertible, the inverse of A will be denoted by A Ex1 Not every matrix is invertible For example, A = is not invertible: a b a b 1 0 =, no matter what a, b, c, d are 0 0 c d [ Ex2 Let A = [ 7 5 Show that C = 3 2 is the inverse of A Question In general, how can we find the inverse of a given matrix? When given matrix is a 2 2 matrix, there is a simple formula a b Theorem 1 Let A = If ad bc = 0, then A is not invertible (or A is singular) If c d ad bc 0, then A is invertible and its inverse is given by A 1 1 d b = ad bc c a Remark The quantity ad bc therefore has an important meaning for A This number is called the determinant of (a 2 2 matrix) A and denoted by det A 2 5 Ex3 Find the inverse of A = 3 7
4 4 Ex4 Find the inverse of B = [ Ex5 The matrix [ [ 1 4 is not invertible, because det 3 12 Question Why is the inverse important? One reason is this: suppose we are to solve a system A x = b (Matrix-vector representation of a system) To obtain the solution set, we have been looking for the rref of the augmented matrix associated with the system If A is an invertible square (say n n) matrix, then there is another method to find the solution set Consider the vector p = A 1 b Then A p = A(A 1 b) = (AA 1 ) b = I n b = b, which means p is a solution of the system We claim that this is the only solution of the system Indeed, suppose w is a solution of A x = b, that is, A w = b Multiplying both sides by A 1 on the left gives w = A 1 b = p Thus we obtained the following Theorem 2 If A is an invertible n n matrix, then for each b in R n, the system A x = b has the unique solution A 1 b = Ex6 Solve the system { 2x1 +5x 2 = 2 3x 1 7x 2 = 1 Method 1 (using the rref of the augmented matrix) Method 2 (using the inverse of the coefficient matrix)
5 5 Theorem 3 Let A be an n n square matrix (i) If A is an invertible matrix, then A 1 is also invertible and (A 1 ) 1 = A (ii) If A and B are n n invertible matrices, then so is AB, and (AB) 1 = B 1 A 1 (iii) If A is an invertible matrix, then so is A T, and the inverse of A T is the transpose of A 1 That is, (A T ) 1 = (A 1 ) T We learned in Theorem 1 how to find the inverse of a 2 2 matrix What if the given matrix is not 2 2? [ Theorem 4 Let A be an n n matrix Form an n 2n matrix A I n If the rref of A I n is of the form I n B (which is equivalent to saying that the rref of A is I n ), then A is invertible and A 1 = B If the rref of A I n is not of the form I n B (which is equivalent to saying that the rref of A is not equal to I n ), then A is not invertible Ex7 Find the inverse of A = 1 0 3, if it exists Ex8 Find the inverse of A = [ , if it exists Method 1 (Using the method in Theorem 4) Method 2 (Using the formula for 2 2 matrices)
6 6 Ex9 Let B = Is B invertible? Recall that a 2 2 matrix A is invertible if and only if det A 0 For a general matrix, it would be nice if there are some characterizations of invertibility This is the main subject of the following section
7 7 Sec 23 Characterizations of Invertible Matrices The Invertible Matrix Theorem, 1st version Let A be an n n matrix Then the following statements are equivalent That is, for a given A, the statements are either all true or all false (a) A is invertible (ie, there exists an n n matrix B such that AB = I n = BA) (b) A is row-equivalent to I n (c) A has n pivot positions (d) The equation A x = 0 has only the trivial solution (e) The columns of A are linearly independent (f) The equation A x = b has at least one solution for each b R n (ie, the system A x = b is consistent for each b R n ) (g) The columns of A span R n (h) A is left invertible, ie, there exists an n n matrix C such that CA = I n (i) A is right invertible, ie, there exists an n n matrix D such that AD = I n (j) A T is invertible Remark What is the difference between statements (a) and (h)? Which statement looks stronger? If (a) is true, then clearly (h) is also true - simply take C = B In other words, the statement (a) clearly implies (h) (or (a) is stronger than (h)) How about the converse? The existence of a matrix C such that CA = I n does not seem to guarantee the existence of a matrix B such that BA = I n and AB = I n right away However, The Invertible Matrix Theorem says that the converse is also true Summary: To show that a matrix A is invertible, it is sufficient to show that there is a matrix C such that CA = I n Similarly, if one can show that there is a matrix D such that AD = I n, then this is enough to guarantee the invertibility of A Problem Suppose that A, B are n n matrices and that AB is invertible and B are invertible Show that both A In the following examples, all the matrices are n n square matrices unless stated otherwise Ex1 Can a square matrix with two identical columns be invertible? Why or why not? Ex2 Can a square matrix with two identical rows be invertible? Why or why not?
8 8 Ex3 If the equation G x = b has more than one solution for some b in R n, can the columns of G span R n? Ex4 If the columns of a matrix A are linearly independent, then do the columns of A 2 span R n? Ex5 If the equation H x = c is inconsistent for some c R n, what can you say about the equation H x = 0? Ex6 Suppose that B is invertible and the equation AB x = 0 has a nontrivial solution Show that A has at most n 1 pivots
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