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ASSIGNMENT 2
Course: Mathematics for Engineering
Student Name:
Student Code:
Question 1. Determine whether each statement is true or false.
(a) A linear system of two equation in two variables is always consistent.
(b) A linear system of two equations in three variables is always consistent.
(c) A linear system of three equations in two variables is always consistent.
(d) Every homogeneous system of linear equations is consistent.
(e) If a homogeneous system of linear equations has more variables than equations, then it
has a nontrivial solution (in fact, infinitely many).
(f) If the linear system is homogeneous, every solution is trivial.
(g) If a linear system has n variables and m equations, then the augmented matrix has m
rows and n + 1 columns.
(h) If there exists a trivial solution, the linear system is homogeneous.
(i) If a linear system has a nontrivial solution, it cannot be homogeneous.
Question 2. Find values of a, b, and c such that the following system is consistent



x +2y −z = a


2x +y +3z = b



 x −4y +9z = c
2
.
Question 3. Find (if possible) conditions on a, b, and c such that the following systems has
no solution, one solution, or infinitely many solutions.

 x +by = −1
(a)
 ax +2y = 5



2x +y −z = a


(b)
2y +3z = b



 x
−z = c
Question 4. Find all values of a for which the system has nontrivial solutions, and determine
all solutions.



x +2y +z = 0


x +3y +6z = 0



 2x +3y +az = 0
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Question 5. Solve the following system



x −2x2 +x3 +3x4 = 0

 1
2x1 −3x2
+9x3 = 0



 x
−x2 −x3 +6x4 = 0
1
Question 6. Find the rank of each of the following matrices.




1
1 2
3 −2 1 −2








(a) A =  3 −1 1
(b) B =  1 −1 3 5 




−1 3 4
−1 1 1 −1
4
Question 7. Carry each of the following matrices to reduced row-echelon form.




2 −1 0
1


1 −3 2 −1




(a) A =
(b) B =  1 −1 3
1


2 −6 2 3
−1 2 −9 −2
Question 8. Suppose


2 −1 0




A= 3
0 1 ,


−1 4 1
Compute AB,
2A − 3B T ,
B2.


x 0 y




B = −2 1 0  .


y 0 −x
5
Question 9. Let A, B, C be matrices such that the indicated operations can be performed.
Determine whether each of the following statements is true or false?
(a) If AB = AC, then B = C.
(b) If AB = AC and A is invertible, then B = C.
(c) (AB)−1 = B −1 A−1 ,
(AB)T = B T AT .
(d) For any matrix A, the matrix AAT is symmectric.
(e) If the (i, j)-entry of A is m then (j, i)-entry of AT is m.
(f) (kA + hB)T = kAT + hB T
(k, h ∈ R).
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(g) (cA)−1 = A−1 , for c ̸= 0.
c
(h) If det A = 0, then A has two equal rows.
(i) det(A + B) = det(A) + det(B).
(k) |cA| = cn |A|, |A−1 | = |A|−1 , |cA−1 | = cn |A|−1 , |(cA)−1 | = c−n |A|−1
for any n × n invertible matrix A and c ̸= 0.
Question 10. Let A, B and C be 3 × 3 matrices such that |A| = 4, |B| = −2 and |C| = 2.
Find
|−2A| ,
(5C)−1 ,
4AT B −1 , (2A)−1 (BC)T ,
A−2 B −1 C 4 .
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Question 11. Use determinants to find the values of c make each of the following matrices
invertible


c−1
3

(a) 
2
c−2


1 0 3




(b) 3 −4 c 


2 5 8
Question 12. Compute the determinants of the following matrices.




a
b
c
1 b c








(b) A = a + 1 b + 1 c + 1
(a)  b c 1




a−1 b−1 c−1
c 1 b
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

a b c




Question 13. If det  p q r  = −1, compute


x y z

−2a
−2b
−2c





det 2p + x 2q + y 2r + z 


3x
3y
3z
Question 14. Find A when


1
1
−1

(a) AT + 3I
= 2
−1 3

−1
1 −1

(b) (−3A)T = 
0 2
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Question 15. Find the characteristic polynomial, eigenvalues, eigenvectors of each matrix.




7 0 −4


2 −4




(a) A =
(b) A = 0 5 0 


−1 −1
5 0 −2