1. Let A = [a)ax5, B = [bj]ax5, C = [cy]sx2, D = [dy]ax2, and E = [eg]sx4- Determine which of the following matrix expressions are defined. For those which are defined, give the size of the resulting matrix. (a) BA (b) AC + D (c) AE + B (d) AB + B (e) E(A+ B)
1. Let A = [a)ax5, B = [bj]ax5, C = [cy]sx2, D = [dy]ax2, and E = [eg]sx4- Determine which of the following matrix expressions are defined. For those which are defined, give the size of the resulting matrix. (a) BA (b) AC + D (c) AE + B (d) AB + B (e) E(A+ B)
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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![UNIVE20649237564332Y pdf
2
DEPARTMENT OF MATHEMATICS AND STATISTICS
MM201 Linear Algebra and Differential Equations
1
Exercises: Chapter 1
1. Let A = [a]ax5, B = [byjlax5, C = [cy]5x2, D = [dij]4x2, and E = [eij]5x4- Determine
which of the following matrix expressions are defined. For those which are defined, give the
size of the resulting matrix.
(b) AC + D (c) AE + B
(h) (A" + E)D.
(а) ВА
(d) AB + B (e) E(A+ B)
(f) E(AC) (g) ETA
2. Consider the matrices
3 0
-1 2
A =
B =
C =
6 1 3
-1 1 2
152
D=
-1 0 1
3 2 4
E =
413
Compute the following (where possible).
(a) 4E – 2D (b) 2B – C (c) AB
(d) BA
(e) (3E)D
(f) (AB)C
(g) A(BC) (h) (DA)" (i) ATD" (j) tr(4ET – D)
(k) tr(CC").
3. Solve the following matrix equation for a, b, c and d:
a - b
b+e
3d +e 2a – 4d
[::)-
4. Let
B =
C =
Verify that (AB)-1 = B-' A- and (ABC)- = C- B-'A-!.
MM201 Linear Algebra and Differential Equations
5. Let D be a non-singular matrix such that
-3
(7D)--
Find D.
6. Evaluate tr(0,xn) and tr(1), and prove that
(a) tr(A+ B) = tr(A) + tr(B),
(b) tr(AB) = tr(BA),
(c) tr(AA) = tr(A)
for any n x n matrices A, B and A ER.
7. In each of the following, determine whether the matrix is in echelon form, reduced echelon
form, both or neither.
1EU 3 0
10 05
0 01 3
0!10
1031
(a)
(b)
(c)
0 0 0 0 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff60e802e-f12d-4d81-92dc-184c117563d2%2Fabe65756-1cc5-4bfd-9d58-1d065e61c5f0%2F66k4jxk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:UNIVE20649237564332Y pdf
2
DEPARTMENT OF MATHEMATICS AND STATISTICS
MM201 Linear Algebra and Differential Equations
1
Exercises: Chapter 1
1. Let A = [a]ax5, B = [byjlax5, C = [cy]5x2, D = [dij]4x2, and E = [eij]5x4- Determine
which of the following matrix expressions are defined. For those which are defined, give the
size of the resulting matrix.
(b) AC + D (c) AE + B
(h) (A" + E)D.
(а) ВА
(d) AB + B (e) E(A+ B)
(f) E(AC) (g) ETA
2. Consider the matrices
3 0
-1 2
A =
B =
C =
6 1 3
-1 1 2
152
D=
-1 0 1
3 2 4
E =
413
Compute the following (where possible).
(a) 4E – 2D (b) 2B – C (c) AB
(d) BA
(e) (3E)D
(f) (AB)C
(g) A(BC) (h) (DA)" (i) ATD" (j) tr(4ET – D)
(k) tr(CC").
3. Solve the following matrix equation for a, b, c and d:
a - b
b+e
3d +e 2a – 4d
[::)-
4. Let
B =
C =
Verify that (AB)-1 = B-' A- and (ABC)- = C- B-'A-!.
MM201 Linear Algebra and Differential Equations
5. Let D be a non-singular matrix such that
-3
(7D)--
Find D.
6. Evaluate tr(0,xn) and tr(1), and prove that
(a) tr(A+ B) = tr(A) + tr(B),
(b) tr(AB) = tr(BA),
(c) tr(AA) = tr(A)
for any n x n matrices A, B and A ER.
7. In each of the following, determine whether the matrix is in echelon form, reduced echelon
form, both or neither.
1EU 3 0
10 05
0 01 3
0!10
1031
(a)
(b)
(c)
0 0 0 0 1
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