Complete parts a and b below for the matrix A. -49 63 28 - 35 - 21 21 49 28 - 42 - 49 14 42 35 - 35 A= - 35 49 42 - 35 42 - 14 - 56 21 - 35 - 56 7 49 28 - 56 -42 56 35 - 28 - 28- 63 -21 a. Construct matrices C and N whose columns are bases for Col A and NulA, respectively, and construct a matrix R whose rows form a basis for Row A. c-O N=O R= b. Construct a matrix M whose columns form a basis for Nul A", form the matrices S=[RT N]and T=[c m], and explain why S and T should be square. Verify that both S and T are invertible. M=[ The matrix S= [RT N]is -O because the columns of R"and N are in Rand dim Row A+ dim Nul A=D. The matrix T=[CM]isI×O because the columns of C and M are in R and dim Col A + dim Nul AT =O by the Rank Theorem, since Col A = Row A".
Complete parts a and b below for the matrix A. -49 63 28 - 35 - 21 21 49 28 - 42 - 49 14 42 35 - 35 A= - 35 49 42 - 35 42 - 14 - 56 21 - 35 - 56 7 49 28 - 56 -42 56 35 - 28 - 28- 63 -21 a. Construct matrices C and N whose columns are bases for Col A and NulA, respectively, and construct a matrix R whose rows form a basis for Row A. c-O N=O R= b. Construct a matrix M whose columns form a basis for Nul A", form the matrices S=[RT N]and T=[c m], and explain why S and T should be square. Verify that both S and T are invertible. M=[ The matrix S= [RT N]is -O because the columns of R"and N are in Rand dim Row A+ dim Nul A=D. The matrix T=[CM]isI×O because the columns of C and M are in R and dim Col A + dim Nul AT =O by the Rank Theorem, since Col A = Row A".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Complete parts a and b below for the matrix A.
- 49
63
28 - 35 - 21
21
49
28 - 42 - 49
14
42
35 - 35
A=
- 35
49
42 - 35
42 - 14 - 56
21 - 35 - 56
7
49
28 - 56
- 42
56
35 - 28 - 28 - 63 -21
a. Construct matrices C and N whose columns are bases for Col A and Nul A, respectively, and construct a matrix R whose rows form a basis for Row A.
c=0
C =
N=O
R=
b. Construct a matrix M whose columns form a basis for Nul A', form the matrices S= RT N and T= C M, and explain why S and T should be square. Verify that both S and T are invertible.
M=
The matrix S= RT N isx because the columns of RT and N are in RU and dim Row A + dim Nul A =.
[RT N]#
The matrix T=[C M ]is xO because the columns of C and M are in R and dim Col A + dim Nul A
by the Rank Theorem, since Col A = Row AT.
The columns of the matrix S are linearly
The columns of S form a basis of R
Thus, by the Invertible Matrix Theorem, S is invertible.
The columns of the matrix T are linearly
The columns of T form a basis of RU. Thus, by the Invertible Matrix Theorem, T is invertible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffba142dd-1494-4b78-9e41-2122df5fca48%2Ff264796f-66af-483d-b3c0-9b58498c5aff%2Ft0y2fcd_processed.png&w=3840&q=75)
Transcribed Image Text:Complete parts a and b below for the matrix A.
- 49
63
28 - 35 - 21
21
49
28 - 42 - 49
14
42
35 - 35
A=
- 35
49
42 - 35
42 - 14 - 56
21 - 35 - 56
7
49
28 - 56
- 42
56
35 - 28 - 28 - 63 -21
a. Construct matrices C and N whose columns are bases for Col A and Nul A, respectively, and construct a matrix R whose rows form a basis for Row A.
c=0
C =
N=O
R=
b. Construct a matrix M whose columns form a basis for Nul A', form the matrices S= RT N and T= C M, and explain why S and T should be square. Verify that both S and T are invertible.
M=
The matrix S= RT N isx because the columns of RT and N are in RU and dim Row A + dim Nul A =.
[RT N]#
The matrix T=[C M ]is xO because the columns of C and M are in R and dim Col A + dim Nul A
by the Rank Theorem, since Col A = Row AT.
The columns of the matrix S are linearly
The columns of S form a basis of R
Thus, by the Invertible Matrix Theorem, S is invertible.
The columns of the matrix T are linearly
The columns of T form a basis of RU. Thus, by the Invertible Matrix Theorem, T is invertible.
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