LinAlgForCal2v4 - MATRICES AND LINEAR ALGEBRA: A QUICK...

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Unformatted text preview: MATRICES AND LINEAR ALGEBRA: A QUICK INTRODUCTION 1. An Introduction to Matrices 1.1. Definition of a matrix. Definition 1.1. An m n matrix is a rectangular array of numbers with m rows and n columns. Example 1.2. A 2 1 matrix is a column vector and a 1 2 matrix is a row vector. Example 1.3. 2 2 matrices. A = 3 4- 1 is an example of a 2 2 matrix. We sometimes write A = ( a jk ) where a jk is the element in the j th row and k th column of A . For example a 12 = 4. Definition 1.4. We say that two matrices A and B are equal if a jk = b jk for all j and k . Example 1.5. 3 3 matrices. B = 4 2 1- 3- 1 3- 2 is a 3 3 matrix. Example 1.6. C = 4 2- 1 1 e 45 is a 3 2 matrix. c 12 = 2 while c 21 =- 1. 1.2. Matrix Addition. Matrices of the same size can be added, and the addition is per- formed component-wise. Example 1.7. 2 3 1- 3 6 + - 2 5 3- 2 4 = 8 3 1- 5 10 Example 1.8. 3 2 1 4 + 4 2 1- 2 1 0 cannot be added! They are different sizes! Symbolically, if A = ( a jk ) and B = ( b jk ) are both m n matrices, then A + B = ( a jk + b jk ). 1.3. Multiplication of a matrix by a scalar. If R is a scalar and A = ( a jk ) is an m n matrix, then the matrix A = ( a jk ). Example 1.9. If A = 4 2 3 3- 3 8 2 , then 1 2 A = 2 1 3 2 3 2- 3 2 4 1 . 1 1.4. A special class of matrices. Definition 1.10. An n n matrix is called a square matrix . Example 1.11. A = 3 9 8 1 3 0 0 0 3 is a 3 3 square matrix. A = 3 9 8 1 3 0 is a 2 3 matrix, and hence not square. Definition 1.12. The identity matrix is an n n matrix with 1s on the diagonal and 0s off of the diagonal and denoted by I or I n . Example 1.13. I 2 = 1 0 0 1 . A zero matrix is any matrix that contains only 0s. Theorem 1.14. Let A,B,C be m n and , R be scalars. Then the following algebraic rules hold: (i) A + B = B + A Commutativity of Matrix Addition (ii) ( A + B ) + C = A + ( B + C ) Associativity of Matrix Addition (iii) ( A + B ) = A + B (iv) ( + ) A = A + A (v) ( ) A = ( A ) 1.5. Problems. 1. Give an example of a 4 2 matrix and a 2 4 matrix. 2. If M = 4 3 1- 1 2- 4 3 1 , then find m 23 and m 22 + m 13 . Let A = 2 3- 1 3 , B =- 1 0 3 8 , and C = 1- 1 3 , 3. What is b 11- 3 a 22 + 2 c 21 ? 4. Compute A + B . 5. Compute 3 C . 6. Compute A + 4 B . 7. Compute 3 B- 2 C . 8. Give an example two matrices F and G so that F + G does not exist. 2. Determinants of 2 2 and 3 3 matrices Definition 2.1. Let A = a b c d . The determinant of A , denoted det( A ) or | A | is the quantity a b c d = ad- bc....
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LinAlgForCal2v4 - MATRICES AND LINEAR ALGEBRA: A QUICK...

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