MTH 100 : Worksheet 2
1. (a) Row reduce the augmented matrix of the system given below to an
RREF matrix using elementary row operations.
3x + 2y + 7z + 9w = 7
6x + 14y + 22z + 15w = 13
x + 4y + 5z + 2w = 2
(b) Is the system consistent or inconsistent? If consistent, express the solution in the form of a vector u which is a solution of the non-homogeneous
system plus scalar multiples of vector(s) which are solutions of the associated homogeneous system.
2. (a) Row reduce the augmented matrix of the system given below to an
RREF matrix using elementary row operations.
x + 5y − 3z = −4
−x − 4y + z = 3
−2x − 7y = a
(b) For what values of a is the above system consistent and for what values
of a is it inconsistent? Justify your answer.
3. For what values of a and b, the system
x+y+z = 3
2x + 3y + 4z = 9
x − y + az = b
has
(a) Unique solution
(b) Infinitely many solutions
(c) No solution
4. Is it possible for a non-homogeneous system Ax = b, b ̸= 0, to be inconsistent when the associated homogeneous system Ax = 0 has a unique
solution (i.e. only the trivial solution)? Answer YES or NO, and justify
your answer. If YES, construct an example and verify. If NO, explain
with reference to suitable propositions and theorems.
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5. (a) Find the values of x for which the following matrix is an augmented
matrix corresponding to a consistent system.


1 −2
1
x
5
−2 x2 
A = 0
4 −23 10 x3
(b) Find the RREF of the matrix formed by replacing x in A by π.
6. Prove that : If the matrix B has been obtained from the matrix A by an
elementary row operation, then the vector v is a solution of the homogeneous system Ax = 0 if and only if v is a solution of the homogeneous
system Bx = 0
(Note: Think about a similar version for a non-homogeneous system and
prove it.)
7. Determine the inverse of the given matrix A using row reduction.


2 1 −1
A = 0 2 1 
5 2 −3
8. TRUE OR FALSE?
(a) The sum of two invertible matrices(square matrices of same order) is
always invertible.
(b) If matrices A and B commute ,i.e. AB = BA, then invertibility of A
implies invertibility of B.
Justify your answer. Prove if TRUE or give counter-example if FALSE.
9. Suppose AB = AC,where B and C are n × k matrices and A is invertible.
Show that B = C. Is this true in general when A is not invertible? Justify
your answer (prove if true, give counter-example if false).
10. (a) Show that an elementary matrix E obtained by replacement of a row
Ri of I by Ri + kRj , where j < i, is a unit lower triangular matrix.
(b) Show that the product of two unit lower triangular matrices is again
a unit lower triangular matrix.
(c) Show that if A is a unit lower triangular matrix, then A is invertible
and A−1 is also a unit lower triangular matrix.
11. For each of the following clearly state True or False (prove if true, counter
example if false)
• For any square matrix A, if Ak is invertible for some positive integer
k > 1, then A itself is invertible.
• If a 3 × 3 square matrix A satisfies A3 = 0, then A = 0.(Here 0
indicates the zero matrix.)
2

1
0
−1
12. Check whether A is invertible and find A if it exists. A = 
0
0
2
2
0
0
3
3
3
0

4
4

4
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13. Suppose A is 2 × 1 matrix and B is 1 × 2 matrix.Prove that C = AB is
not invertible.
14. Prove the following generalization of previous problem.
If A is m × n matrix and B is n × m matrix and n < m then prove that
AB is not invertible.
15. Let A be an n × n (square) matrix.Prove the following statements:
• If A is invertible and AB = 0 for some n × n matrix B, then B = 0.
• If A is not invertible then there exists an n × n matrix B such that
AB = 0, but B ̸= 0
a b
16. Let A =
Prove using elementary row operations that A is invertible
c d
iff ad − bc ̸= 0
17. Prove that an upper triangular(square) matrix is invertible iff every entry
on the main diagonal is different from zero.(An n × n matrix A = [aij ]n×n
is called upper triangular if aij = 0 for i > j ,i.e. every entry below the
main diagonal is zero.)
18. Given an m × n matrix
A and n × k matrix B, the product AB =
Av1 Av2 .... Avk in column form where B = v1 v2 .... vk is
in column form. Construct an example to illustrate this rule. The matrix
A in your example should be atleast 3 × 3 and B should be atleast 3 × 2.
19. Prove the following proposition in general case,i.e. for any row operation
e and any matrix A.
Proposition If e is an elementary row operation and E is the m × m
elementary matrix e(Im ), then for every m × n matrix A, e(A) = EA.
(NB: the three cases of scaling, replacement and interchange require separate proofs.)
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