Linear Algebra Exercise 2021
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Linear Algebra Exercise 1 (chapter 1.1-1.3)
Part I For each statement that follows, answer true (T) if the statement is always true and false (F) otherwise.
1. ( ) If a linear system has no solution, we say that the system is inconsistent
2. ( ) Linear equation system Ax = 0 must be consistent.
3. ( ) Two systems of equations involving the same variables are said to be equivalent if they have the same solution set.
4. ( ) An n × n matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB = I .
5. ( ) If A and B are n × n matrices, then A + B = B + A .
6. ( ) If A and B are n × n matrices, then (A 一 B)2 = A2 一 2AB + B2 .
7. ( ) A linear system is said to be under-determined if there are more equations than unknowns.
8. ( ) Over-determined systems are always inconsistent.
Part II Chose the right answer to each question.
1. ( ) Which of the following matrix is in row echelon form?
A = ╱ 0(2) 1(3) ( 0 0 |
3(2) 、 4 . |
B = ╱ 0(0) 1(0) |
0 |
C = ╱ 0(1) 1(0) |
3 3 |
4 |
D = ╱ 0(1) 0(2) ( 0 0 |
0(3) 、 1 . |
2. ( ) Which of the following matrix is in reduced row echelon form?
A = ╱ 、 B = ╱ 、 C = ╱ 、 D = ╱ 0(1) 0(0)
( 0 0 0 . ( 0 2 4 . ( 0 0 1 5 . ( 0 0
3. ( ) Let A be a 3 × 3 matrix and suppose that
3a1 + 2a2 一 5a3 = 0,
how many solutions will the system Ax = 0 have?
A. 0 B. 1 C. 2 D. infinitely many
Part III Find solutions to the following questions.
1. Use Gaussian reduction to solve the following system
一x1 一 2x2 + 3x3 = 1
3x1 一 7x2 + 4x3 = 10
2. A = ╱ 3一2 ( |
1 0 2 |
1(4) 、 2 . |
B = ╱ 1一3 ( 2 |
0 1 一4 |
1(2) 、 1 . |
Compute 2A, A + B, AB and BT AT
3. Consider a linear system whose augmented matrix is of the form
╱ 11(5) 、
( 1 3 a b .
(a) For what values of a and b will the system have infinitely many solutions?
(b) For what values of a and b will the system be inconsistent?
Linear Algebra Exercise 2 (chapter 1.4-1.6)
Part I For each statement that follows, answer true (T) if the statement is always true and false (F) otherwise.
1. ( ) An n x n matrix A is nonsingular if and only if the reduced row echelon form of A is I (the identity matrix).
2. ( ) If A is nonsingular, then A can be factored into a product of elementary matrices.
3. ( ) If A and B are nonsingular n x n matrices, then A + B is also nonsingular and (A + B)− 1 = A− 1 + B − 1 .
4. ( ) If A is a 4 x 4 matrix and a1 + 2a2 = a3 + a4 , then A must be singular.
5. ( ) If A and B are nonsingular n x n matrices, then AB is also nonsingular and (AB)− 1 = B − 1 A− 1 .
6. ( ) If A and B are n x n matrices, then A + B = B + A .
7. ( ) If A and B are n x n matrices, then (A - B)2 = A2 - 2AB + B2 .
8. ( ) If A and B are n x n matrices, then (A + B)T = AT + BT
9. ( ) If A and B are n x n matrices, then (AB)T = AT BT .
10. ( ) If E is an elementary matrix, then ET is also an elementary matrix.
Part II Chose the right answer to each question. 1. ( ) For the following pair of matrix, find an elementary matrix B = ╱ 2(1) ( 1 E equals (A) E1 = ╱ 0(0) . (C) E3 = ╱ 0(1) ( 0 |
E 0 5 2
╱ ( ╱ ( |
such that EA = B 3(8) 、 3 .
|
2. ( ) For the following pair of matrix, find an elementary matrix E such that EA = B
A = ╱ 、 B = ╱ 、
( 1 0 8 . ( 1 0 8 .
E equals
(A) E1 = .(、) (B) E2 = .(、)
(C) E3 = 0(1) 2(0) 0(0) .(、) (D) E4 = 0(2) 1(0) 0(0) .(、)
3. ( ) For the following pair of matrices, find an elementary matrix E such that EA = B
E equals
A. ╱ 0(1) ( 2
A = ╱ 2(1)
( 1
0 1 0 |
0(0) 、 1 . |
B . ╱ 0(1) 2(0) ( 0 0 |
0(0) 、 1 . |
B = ╱ 2(1) 3(1)
( 3 4
C. ╱ 0(0) 1(0) ( 1 0 |
0(1) 、 0 . |
D . ╱ 0(2) 1(0) ( 0 0 |
0(0) 、 1 . |
Part III Find solutions to the following questions.
1. Let A and B be symmetric n x n matrices. Determine whether the given matrices must be symmetric or could be nonsymmetric:
(a) C = A2
(b) D = ABA
2. Prove that if A is nonsingular then AT is nonsingular and
(AT)− 1 = (A− 1 )T .
3. Find the inverse of matrix A A = ╱ -21 ( |
-3 6 8 |
-13 、 3 . |
4. Let
A = ╱ 3(5) 2(3) 、 B = ╱ 2(6) 4(2) 、 C =
Solve the following matrix equation
AX + B = X
5. Let A be a nonsingular n x n matrix. Calculate
╱ A1 、╱ A I 、
6. Let ╱ 2(5) 2 A = .(.) 1 0 . 0 1 |
0 0 8 5 |
0(0) 、. 2 . |
╱ 4(3) 5 B = .(.) 0 0 . 0 0 |
0 0 4 6 |
0(0) 、. 2 . |
Linear Algebra Exercise 3 (chapter 2.1-2.3)
Part I For each statement that follows, answer true (T) if the statement is always true and false (F) otherwise.
1. ( ) If A and B are n × n matrices, then (A _ B)(A + B) = A2 _ B2 .
2. ( ) If A and B are n × n matrices, then det(A + B) = det(A) + det(B) .
3. ( ) If A and B are n × n matrices, then det(AB) = det(BA) .
4. ( ) If A and B are n × n matrices, then det(A + B) = det(A) + det(B) .
5. ( ) If A is an n × n matrices, then det(cA) = cn . det(A) .
6. ( ) If A is an n × n matrices, then det(cA) = c . det(A) .
7. ( ) If A and B are nonsingular n × n matrices, then A _ B is also nonsingular and (A _ B)− 1 = A− 1 _ B − 1 .
8. ( ) If A is a nonsingular n × n matrix, and det(A) = 5, then det(A− 1 ) = .
9. ( ) A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.
10. ( ) A and B are n × n matrices, det(A) = det(B) implies A = B .
2022-06-30