CALCULUS REV1 S ITED
PART 2
A Self-study Course STUDY GUIDE
Block 4
Matrix Algebra
Herbert I. Gross Senior Lecturer Center for Advanced Engineering Study Massachusetts Institute of Technology Copyright
@
1972 by
Massachusetts Institute of Technology Cambridge, Massachusetts All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the Center for Advanced Engineering Study, M.I.T. CONTENTS Study Guide lock 4 : Matrix Algebra
Pretest
Unit 1:
Unit 2:
Unit 3 :
Unit 4 :
Unit 5:
Unit 6 :
Unit 7 :
Quiz
Linear Equations and Introduction to Matrices
Introduction to Matrix Algebra
Inverse Matrices
Matrices as Linear Functions
The Total Differential Revisited
The Jacobian
Maxima/Minima for Functions of Several Variables
Solutions Block 4 : Matrix Algebra
Pretest
Unit 1:
Unit 2:
Unit 3:
Unit 4 :
Unit 5:
Unit 6 :
Unit 7 :
Quiz
Linear Equations and Introduction to Matrices
Introduction to Matrix Algebra
Inverse Matrices
Matrices as Linear Functions
The Total Differential Revisited
The Jacobian
Maxima/Minima for Functions of Several Variables
Study Guide
BLOCK 4 :
MATRIX ALGEBRA
Study Guide
Block 4 : M a t r i x Algebra
Pretest
1. Solve t h e m a t r i x e q u a t i o n AX
2. Find A-'
A = [
-
BC = 0 i f
;
if
3. Consider t h e system of e q u a t i o n s
How must b3 and b 4 be r e l a t e d t o bl and b2 f o r t h i s system t o
have s o l u t i o n s ?
4 . U s e l i n e a r approximations t o e s t i m a t e t h e p o i n t ( x , y ) n e a r ( 3 . 2 )
f o r which
5 . L e t x be determined a s a f u n c t i o n of z by t h e p a i r of e q u a t i o n s .
Compute
dx
dz -
.
6. Find t h e maximum and minimum v a l u e s o f f (x, y z) = x 2 + y2 + z2
s u b j e c t t o t h e p a i r of c o n s t r a i n t s t h a t x 2 + 2y2 + z2 = 1 and
Study Guide Block 4: Matrix Algebra Unit 1: Linear Equations and Introduction to Matrices 1.
Lecture 4.010 Study Guide Block 4: Matrix Algebra Unit 1: Linear Equations and Introduction to Matrices 2.
Read Supplementary Notes, Chapter 6, Sections A, B, and C. 3.
Exercises: 4.1.1 Compute the matrix product and use this result to show how to express zl and z2 in terms of xl, x2, x3, and x4 if z1 = Y1
2y2
+
3 ~ 3
2 = 3yl + Y2
+
2 ~ 3
+
and
y1 = 3x1 + 6x2
+ 2x3 + 7x4
y2 = 4~1 + 5x2 + x3 + 8x4
y3
=
4x1 + 7x2 + 9x3 + 5x4
4.1.2 Compute Then interpret these two products in terms of systems of linear equations. Study Guide Block 4: Matrix Algebra Unit 1: Linear Equations and Introduction to Matrices 4.1.3 Compute and try to generalize these results. Compute 1
1
2
(
23
4
3
5
2)
(I: -!) (I! -:)
and
1
1
2 (23
3
4
2)
5
Interpret these products in terms of systems of linear equations. Compute and try to generalize this result. 4.1.6 a.
Compute (continued on next page) 4.1.4
Study Guide Block 4: Matrix Algebra Unit 1: Linear Equations and Introduction to Matrices 4.1.6 continued b. (Optional) Let A = (aij) be an n x n matrix and let Eij (i # j)
denote the n x n matrix each of whose elements on the main diagonal and in the ith row, jth column are 1, and everywhere else are
0. Describe the products Eij A.
a. Compute b. (Optional) With A and Eij as in 4.1.6, compute A Eij. 4.1.8 Compute and generalize this result to show how we may multiply each element of a matrix by the same scalar. MIT OpenCourseWare
http://ocw.mit.edu
Resource: Calculus Revisited: Multivariable Calculus
Prof. Herbert Gross
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