=========================preview======================
(math152)[2005](s)midterm~kqyan^_10471.pdf
Back to MATH152 Login to download
======================================================
MATH 152 Applied Linear Algebra and Di.erential Equations Spring 2005-06
2 Review Notes for Linear Algebra C True or False Last Updated: April 28, 2006
The following answers are not guaranteed to be correct. You are welcome to ask if you have any questions.
Chapter 5 [ Matrix ]
5.1 If A is a square matrix and A2 = I, then A = I or A = .I.
.01C
False. A =.
10
5.2 If AB = O, then A = O or B = O.
2003
.100C
False. A =, B =0 0
000 45.
01
5.3 If A, B, C are square and ABC = O, then one of them is O.
.00C
False. A = B = C =.
10
5.4 If AB = AC, then B = C.
False. Choose B =.C and A = O.
5.5 If A is nonzero and AB = AC, then B = C.
.10C.00C.00C
False. A =, B =, C =.
00 00 01
5.6 The square of a nonzero square matrix must be a nonzero matrix.
.01C2 .00C
False. =.
00 00
5.7 If AB = BA, then (A + B)3 = A3 +3A2B +3AB2 + B3 .
True.(A + B)3 =(A + B)(A2 + AB + BA + B2)= A3 + A2B + ABA + AB2 + BA2 + BAB + B2A + B3 = A3 + A2B + A2B + AB2 + A2B + AB2 + AB2 + B3 = A3 +3A2B +3AB2 + B3 .
5.8 An invertible matrix must be a square matrix.
True. A is invertible .. AB = BA = I for some B. If A is m n and B is n m, then we must have m = n.
5.9 A non-square matrix can never be invertible.
True. Equivalent to 5.8.
5.10 If A has a zero row or a zero column, then A is not invertible.
True. A has zero determinant and hence not invertible.
5.11 If A is a square matrix which has no zero rows, then A is invertible.
False. A may have a zero column. Then A is not invertible.
5.12 Let A, B be invertible matrices of same size. Then AB is also invertible.
True. AB is de.ned and square. .
det (AB) = (det A)(det B) = 0.
5.13 Let A, B be invertible matrices of same size. Then A + B is also invertible.
False. Let A invertible. Choose B = .A so that B is invertible. But then A + B = O is not invertible.
5.14 If AB is equal to the identity matrix, then A must be an invertible matrix.
C
, B =
24 35
10
00.100 False. Choose A, B non-square such as A = .001 01
5.15 A, B are square matrices. If AB = I, then BA = I. Hence, A is invertible.
True. We prove it by contradiction. In the following proof,
we need so-called elementary matrices Ei, for instance, like
24 35
,
24 35
,
24 35
. In this case, these 100
1 0 0
0 0 1
0 1 0
0
.6
0 1 00
0 10
001
.4
01
three matrices correspond to the elementary row operations R2 . R3, .6R2, .4R1 + R3, respectively. In general, each row operation always has a corresponding elementary matrix which is also invertible. So, if A is row equivalent to B by doing some row operations to A, then we also mean B = Es E2E1A for some invertible E1, E2, , Es.
Proof: Suppose the square matrix A is not invertible. Then A is not row equivalent to I =. the row echelon form of A must have a zero row =. Es E2E1A has a zero row =. Es E2E1AB = Es E2E1 = invertible ma