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Matrices For grade 1, undergraduate students Made by Department of Math. ,Anqing Teachers college 1 Some notations Definition. A rectangular array of numbers composed of m rows and n columns a11 a 21 A am1 a12 a22 am 2 ... a1n ... a2 n ... amn is called an m n matrix (read m by n matrix). We also say that the matrix A is of, or has, size m n . The elements ai1 , ai 2 , ain form the i-th row of A , and the elements a1 j , a2 j ,..., a mj form the j-th column of A. We will often write A (aij ) mn for A. Definition. If A (aij ) and B (bij ) are m n matrices, then A B aij bij for i=1,2…, m and j=1,…,n. iff .Matrix opertions Definition. If A (aij ) and B (bij ) are two m n matrices, their sum A+B , is the matrix C (c ) ij , where cij aij bij , i=1,2…, m , j=1,2…,n. Definition. If A (aij ) is an m n matrix and r is a number then rA, the scalar multiple of A by r, is the matrix where C (cij ) cij raij , i=1,2…, m and j=1,…,n. .Some properties Proposition 1. The matrices of size m n form a vector space under the operations of matrix addition and scalar multiplication. We denote this vector space by Mmn. The dimension of the vector space Mmn is not hard to compute. We take our lead from the method we used to show that dim Rn=n. Introduce the m nmatrix Ers (eij ) by the requirement 0 if i r , j s, eij 1 if i r , j s. Proposition 2. The vectors Ers | r 1, 2,..., m, s 1, 2,..., n form a basis for Mmn . Therefore dim M mn mn EXAMPLE 1. 3 1 2 0 2 1 3 0 22 2 7 4 0 1 3 1 2 9 4 0 7 6 1 1 1 4 3 3 0 7 0 9 1 4 4 9 3 . 1 6 0 1 5 5 1 EXAMPLE 2. 1 4 6 0 1 0 1 1 4 7 2 0 1 7 9 1 2 3 7 9 1 0 4 1 6 1 0 4 1 7 2 1 0 2 1 3 7 7 9 9 1 5 5 4 8 1 2 4 14 18 2 Matrix products Definition. If A (aij ) is an m n matrix n p and B (bij ) is an matrix, their matrix product A B is the m p matrix, where AB (cij ) n cij aik bkj i 1,..., m, j 1,..., p. k 1 Remark. Note that for the product of A and B to be defined the number of columns of A must be equal to the number of rows of B. Thus the order in which the product of A and B is taken is very important, for AB can be defined without AB being defined. EXAMPLE 4. Compute the matrix product 4 (1 2 3) 5 . 6 Solution. Note the answer is a matrix 4 (1 2 3) 5 (4 10 18) (32). 6 Remark. Note that the product 4 5 (1 2 3) 6 is not defined. EXAMPLE 5. Compute the matrix product 0 1 1 0 1 1 0 0 1 0 0 1 . 0 0 0 0 0 0 Answer . 0 0 1 0 0 0 0 0 0 Definition. A matrix A is said to be a square matrix of size n iff it has n rows and n columns (that is the number of rows equals the number of columns equals n). Remark. It is easy to see that if A and B are square matrices of size n then the products AB and BA are both defined. However they may not be equal.. EXAMPLE 7. Let 1 0 3 0 A and B 0 3 2 1 Compute the matrix products AB and BA. Solution. We have 1 0 3 0 3 0 3 0 1 0 3 0 AB , BA 0 3 2 1 6 3 2 1 0 3 2 3 and so we see that AB BA. Remark. As the preceding example shows even if AB and BA are defined we should not expect that AB=BA. Notation. If A is a square matrix then AA is defined and is denoted by A2. Similarly, A... A n times is defined and denoted by A . n EXAMPLE 8. Let 0 0 A . 1 0 Calculate A2 . Solution. We have 0 0 0 0 0 0 A 1 0 1 0 0 0 2 .The rules of matrix operations (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) A+B=B+A A+(B+C)=A+(B+C) r(A+B)=rA+rB A+0=A 0A=0 A+(-1)A=0 (r+s)A=rA+sA (A+B)·C=A·C+C·B 0·A=0=A·0 A·(B·C)=(A·B) ·C 3 Special types of matrices Diagonal matrices. a11 A 0 0 a nn Triangular matrices. A square matrix A is said to be lower triangular iff A= ( aij ) where aij 0 if j i For example 1 0 0 0 2 0 3 1 3 is a lower triangular matrix. The Zero matrix. The zero matrix is the matrix 0 all of those entries are 0. Idempotent matrices. A square matrix A is said to be idempotent iff A2 A Nilpotent matrices. A square matrix A is said to be nilpotent iff there is an integer q such Aq 0 . Nonsingular matrices. A square matrix A is said to be invertible or nonsingular iff there exists a matrix B such that AB=I and BA=I. 1 A Denoted by . For example if then 0 1 0 A 1 0 0 0 0 1 0 1 0 A 1 1 0 0 . 0 0 1 A nilpotent matrix is not invertible. For suppose that A is a nilpotent matrix that is invertible. Let B be an inverse for A. Since A is nilpotent there is an integer q such that A 0. q Then 0 A B A q so q 1 q 1 AB A I A q 1 Aq 1 0. If we repeat this trick q-1 times we will get A 0. But then I AB 0 B 0, which is impossible. Symmetric and skew-symmetric matrices. A square matrix A= aij is said to symmetric iff aij a ji for it is said to be skew-symmetric iff for aij -a ji , i, j 1, 2,..., n. For example 1 0 1 0 0 0 and 1 1 3 0 1 2 3 1 2 3 4 5 6 5 7 8 6 8 9 are symmetric matrices, and 1 1 2 0 1 1 0 3 and 1 0 -2 3 0 are skew-symmetric matrices. Proposition 3 A 2 2 matrix a b A c d is nonsingular iff ad bc 0. If ad bc 0, then 1 d b A ad bc c a 1 PROOF. Suppose that ad Let 1 d b B ad bc c a Then bc 0. 1 d b a b BA c a c d ad bc 1 da bc bd bd ad bc ca ac ca ad 0 1 ad bc 0 ad bc ad bc 1 0 I. 0 0 and therefore A is nonsingular with 1 d b A . ad bc c a 1 Suppose conversely that A is nonsingular, but that ad bc 0 . We will deduce a contradiction. Let d b C . c a Then computing as above 0 ad bc CA (ad bc) I 0. ad bc 0 This gives the equation 1 1 1 C CI C ( AA ) (CA) A 0 A 0. Therefore d b 0 0 C . c a 0 0 So that a 0, b 0, c 0, d 0. But then A=0 also, so 1 1 I AA 0 A 0. So 1 0 0 0 0 1 0 0 and hence 1=0, which is impossible. 4 SOME EXERCISES 1. Perform the multiplications following 0 1 0 1 0 1 1 0 0 1 , 0 0 2 1 0 1 0 1 0 1 00 1 2 2 , 0 1 0 1 1 0 2 2 0 1 0 0 0 1 2 3 1 0 0 4 5 6 . 0 0 0 7 8 9 matrix 2. Which of the following matrices are nonsingular, idempotent, nilpotent, symmetric, or skew-symmetric? 1 1 A , 0 0 1 1 B , 1 1 1 1 C , 1 1 1 1 D , 1 1 0 1 F , 1 0 1 0 G , 0 1 1 0 H , 1 0 4 0 J . 0 2 3. If A is an idempotent square matrix show I-2A is invertible (Hint: Idempotent correspond to projections. Interpret I-2A as a reflection. Try the 2 2 case first. Then try to generalize.) Thanks!!!