Unit 1: Algebra

Chapter 1: matrix and determinants

Matrix
Matrices is a rectangular arrangement of numbers in row and column enclosed in

small or big bracket e.g. |10 01| 2×2 matrix.

Size or order of matrix

Number of row and column in the matrix is called size of matrix.

e.g. A= (15 26) then size = 2×3 matrix.

[ ]
4 5 7
C= 8 2 3 Then size =3×3 matrix
4 5 6

Theory type questions

1. State the condition under which two matrices can be added and
multiplied. Answer: - addition of two matrices is hold if of the matrices are
of same order.
While the multiplication of two matrices A and B is possible only if the
number of columns of first matrix must be equal to number of rows to
second. (i.e. 3 ×2 2 ×3)
2. When does the matrix have its inverse? Also, state the condition.
Answer: - Two non-singular matrices A and B will be inverse of each other
if; AB=BA=I.

Some special type of matrix

1. Square matrix: - A matrix having same number of rows and columns is a
square matrix.

( )
2 4 5
For example, A= 3 2 1
4 6 5
2. Unit matrix: - a diagonal matrix, in which all elements in the diagonal are
1, is a unit matrix. Which is also called identity matrix. For example

[ ]
1 0 0
I= 0 1 0 is aunit matrix of order3.
0 0 1
1 0
I=⌊ ⌋ is a unit matrix of order 2.
0 1
Diagonal matrix: - a square matrix in which all elements except in the
diagonal are zero, is a diagonal matrix.
 ( )
5 0 0
For example; A= 0 2 0 is a diagonal matrix.
0 0 −8
3. Null matrix: - a matrix, whose all elements are zero, is a null matrix or a
zero matrix. For example;

( )
0 0 0
O= 0 0 0
0 0 0
4. Singular matrix: - Those square matrixes whose determinants is zero is
known as singular matrix.
5. Non- singular matrix: - Those square matrixes whose determinant is not
zero is called singular matrix. Or those square matrices whose
determinants are non-zero is called non- singular matrix.
6. Transpose of matrix: - if the rows are converted into column and the
column are converted into rows or vice- verse is termed as transpose of
matrix. And is denoted by AT
Exercise

[ ]
2 3
1. If A = [
1 −2 3
−1 2 1 ]
and B= 3 2 , find the matrix AB – 3I, where I is the
1 2
unit matrix of order 2.
2. Construct a 3 ×3 matrix whose a ij are given by;
a. a ij=−2i+3 j
b. a ij=2ij
c. a ij=2i+ j

3. If A= ( 24 −11 ), prove that: A − A−6 I =0.

2

X =(
−1 1 )
4 2
4. , show that ( X −2 I ) ( X−3 I )=0, where I and O are unit matrix and
zero matrix of order 2.

5. D= ( 40 05), find a matrix X such that DX=(12 24).

6. Define a triangular matrix. How would you distinguish between upper and
lower triangular matrixes?
Answer; a square matrix is called a triangular matrix if its entries below or

( )( )
a b c
1 2
above the leading diagonal are zero. For example, , 0 d e
0 3
0 0 f
Upper triangular matrix: - a square matrix in which all the elements below
the leading diagonal are zero is called an upper triangular matrix. For

( )( )
a b c
1 2
example; , 0 d e
0 3
0 0 f
 Lower triangular matrix: - a square matrix in which all the elements above
the leading diagonal are zero is called a lower triangular matrix. For

( )( )
a 0 0
1 0
example; , b c 0 is a lower triangular matrix of order 3×3.
2 3
d e f

7. Let A= ( ) and B = (
1 4)
1 3 3 2
find the transpose matrix of AB.
2 4

| |
1+ x 1 1
8. Evaluate; 1 1+ y 1
1 1 1+ z

9. If C= ( )
1 2
3 1
, show that C2-2C-5I=0.

10. If A= ( ) (
1 2
3 4
, B=
1 0
2 −3
∧C= )
1 −1
0 1 ( )
verify that: A(BC) = (AB)C

2 1 0 2 3
11.Find 5AB if A= ⌊ ⌋ ande B=⌊ ⌋
−3 0 1 −1 0
12.If A =
3 2
1 5 ( )
show that A2-2A-5I=0, where I is the unit matrix of order of

2.

13.If A = (13 21), find A 2

(−1 )
2 3
14.if A=(
3 −7 1 )
4 2 −1
∧B −3 0 ,find the product of AB and BA. Comment on
5
the results.

15.If A= (50 2 −1
−3 2
∧B=)−1 2 3
5 0 −2 ( )
, find the matrix C such that C= 2A+3B.

( ) ( )
0 1 2 1 0 2
16.If A= 1 2 3 ∧B= 0 1 2 ; from the products AB and BA and show that
2 3 4 1 2 0
AB is not equal to BA.

17.Define a matrix. If P= [−14 21] ,. Show that P2-5P+6I=0, Where I and O are

the 2*2 identity and null matrices respectively.

18.If X= prove that x2-4X-5I=0 where I and O are the 3*3 unit and null
matrices respectively.

19.If A=(13 24 ) ,B=(12 −30 ) and C= (10 −11 ) verify that A(BC)=(AB)C
20.If A=(
1 3)
∧B=(
3 2)
2 0 −2 1 T T T
, show that (AB) = B A

| |
3 2 6
3. Define singular matrix. Is the given matrix singular 1 1 2 ?
2 2 5
Exam pattern Question
1. Given the system 3x−y=8and 6x+2y=5:
a. Mention two methods to solve the system using matrices. [1]
b. Define what it means for a system to be inconsistent. [1]
c. Write the expressions for x and y by applying Cramer’s rule. [2]
d. Discuss the implication of having a non-zero determinant for the
system.
2. Given the system of equations 2x+3y+1=0 and −x+4y−2=0:
a. Write the system in matrix form, AX=BA , identifying A, X, and B. [1]
b. Define what is meant by the transpose of a matrix and find the
transpose of A. [2]
c. Describe the steps of the Gaussian elimination method to solve this
system. [2]
d. What does the adjoint of a matrix represent? [1]
3. For the linear equations 3x+5y+2=0 and 7x−y+6=0:
a. Express this system in the form AX=BA, and identify matrix A. [1]
b. Find the transpose of matrix A. [1]
c. Explain the Gaussian elimination process to solve this system. [2]
d. Define the adjoint of a matrix and its use in finding the inverse. [1]
4. Consider the equations ax+by+c=0 and dx+ey+f=0:
a. Formulate these equations in matrix notation, clearly identifying
each matrix. [1]
b. Describe the steps to obtain the adjoint of a 2×2 matrix. [2]
c. Define what is meant by a row-equivalent system in the context of
Gaussian elimination. [1]
d. What is the result of finding the transpose of an adjoint matrix? [1]
4. For the system x+2y−3=0 and 4x−y+5=0:
a. Rewrite the system in matrix form, identifying each component
matrix. [1]
b. Define the transpose of a matrix and find the transpose of the
matrix formed by the coefficients. [2]
c. Outline the Gaussian elimination steps to find the solution for x and
y. [2]
d. Explain what an adjoint matrix is and calculate the adjoint of the
coefficient matrix. [2]
5. Given the system 2x+3y=4 and x−y=1:
a. Arrange this system in the matrix form AX=BA and identify the
matrices A and B. [1]
b. Explain the role of Gaussian elimination in solving linear systems.
[1]
c. Define the transpose and adjoint of a matrix. Calculate both for A.
[3
d. If det(A)≠0, what does this imply about the existence of solutions?
[1]
6. What is skew-symmetric matrix. [1]

[ ]
2 3 −1
a. If A= 0 4 6 , find A+ A T [1]
−2 3 0
b. Is the property A−A T , a skew-symmetric matrix or symmetric
matrix show it [2]
 c. Under which condition the addition of matrix is impossible? Sate
any condition with example [1]
7. Write down the condition for which multiplication of matrix is possible? [1]
a. What is singular matrix? [1]
b. State the identity matrix of order 2*2. [1]
c. Are you well know about sarrus rule, compute the value of the

[ ]
1 2 3
given determinants using this rule, 4 5 6 [2]
7 8 9

Determinants

| |
b+c a−b a
3 3 3
8. c+ a b−c b =3 abc−a −b −c
a+b c−a c

| |
( b +c )2 a2 a2
3
b =2 abc ( a+b +c )
2
9. b ( c +a )2 2

c2 c2 ( a+ b )2

| |
1 a bc
10. 1 b ca =( a−b ) ( b−c ) ( c−a )
1 c ab

| |
1 a a3
11. 1 b b3 = (a-b) (b-c) (c-a) (a +b + c)
3
1 c c

| |
x− y−z 2x 2x
12. 2y y−z −x 2 y = (x + y + z)3
2z 2z z−x− y

| |
1+ a 1 1
13. 1 1+a 1 =¿ a2(a+3)
1 1 1+a

| |
1+ x 1 1
14. 1
1
1+ y
1 1+ z
1 1 1
1 =xyz 1+ + +
x y z ( )
| |
1 1 1
15. a b c = ( a−b ) ( b−c )( c−a ) ( a+b+ c )
3 3 3
a b c

| |
x 2 +1 xy xz
2 2 2 2
16. xy y +1 yz =1+ x + y + z
2
xz yz z +1

| |
1 1 1
17. α β ¥ =( α− β)(β−¥ )(¥ −α )
β¥ α¥ αβ
18. Solve the following equations by using Cramer’s rule
a. 2x + 3y = 7; 3x + 5y = 9
 5x + 3y = 17; 3x + 7 y = 31
b.
c. 2x + y − z = 3, x + y + z = 1, x − 2 y − 3z = 4
d. x + y + z = 6, 2x + 3y − z = 5, 6x − 2 y − 3z = − 7
e. x + 4 y + 3z = 2, 2x − 6 y + 6z = −3, 5x − 2 y + 3z = −5
19. At marina two types of games viz., Horse riding and Quad Bikes
riding are available on hourly rent. Keren and Benita spent ₹780 and
₹560 during the month of May.

20. Find the hourly charges for the two games (rides). (Use Cramer’s
rule).
In a market survey three commodities A, B and C were considered. In finding out
the index number some fixed weights were assigned to the three varieties in each
of the commodities. The table below provides the information regarding the
consumption of three commodities according to the three varieties and also the
total weight received by the commodity

Find the weights assigned to the three varieties by using Cramer’s Rule.
21. A total of ₹8,500 was invested in three interest earning accounts. The interest rates
were 2%, 3% and 6% if the total simple interest for one year was ₹380 and the
amount invested at 6% was equal to the sum of the amounts in the other two
accounts, then how much was invested in each account? (use Cramer’s rule).