Karnataka 2nd PUC Mathematics Matrices MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: MATRICES

Definition:
A matrix is a rectangular arrangement of numbers or functions enclosed within the
brackets.
The numbers or functions are called elements of the matrix.
Matrices are denoted by capital letters 𝐴, 𝐵, 𝐶, … … … … ..
2 3 1
Example: 𝐴 = [ ]
−1 2 3
In the above example, the horizontal lines of elements are called rows and
the vertical lines are called columns.
Order of the Matrix:
A matrix having m rows and n columns is called a matrix of order 𝑚 × 𝑛
2 3 1
Example: 𝐴 = [ ] is a matrix of order 2 × 3
−1 2 3
Note:
 In general matrix of order 𝑚 × 𝑛 is written as 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛
Where 𝑎𝑖𝑗 denotes the element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column.
 Number of elements in a matrix of order 𝑚 × 𝑛 is given by mn
Types of Matrices:
Row matrix:
A matrix having only one row is called row matrix. In general order of row matrix
containing n columns is given by 1 × 𝑛
Example:- 𝐴 = [1 2 3]1×3
Column matrix:
A matrix having only one column is called column matrix. In general order of column
matrix containing n rows is given by 𝑛 × 1
1
Example:- 𝐵 = [2]
3 3×1
2026-27 MATHEMATICS CET MATERIAL Page 1 of 16

Square matrix:
A matrix in which the number of rows is equal to the number of columns (𝑚 = 𝑛)
is called square matrix.
6 2
Example:- 𝐴 = [ ] is the square matrix of order 2.
5 8 2×2
Note:
 The diagonal from top left to the bottom right of a square matrix is
called Principal diagonal.
 The elements of the principal diagonal are called principal diagonal elements.
 If 𝑚 ≠ 𝑛 then matrix is called a Rectangular Matrix.
 The sum of diagonal elements of a square matrix 𝐴 is called the trace of matrix 𝐴
which is denoted by tr𝐴.
1 2 3
Example:- 𝐴 = [−3 3 4] Here 1, 3, −1 are the principal diagonal elements.
3 4 −1
Diagonal matrix:
For any square matrix 𝐴 = [𝑎𝑖𝑗 ], 𝑎𝑖𝑗 = 0, when 𝑖 ≠ 𝑗 is called a diagonal matrix.

1 0 0 
Example:- A = 0 3 0 
 
0 0 -1

Scalar matrix:
For any square matrix 𝐴 = [𝑎𝑖𝑗 ], 𝑎𝑖𝑗 = 0, when 𝑖 ≠ 𝑗, and 𝑎𝑖𝑗 = 𝑘, where 𝑘 is a constant
for𝑖 = 𝑗 is called a scalar matrix.
3 0 0
Example:- A =  0 3 0 
 
 0 0 3 

Unit matrix or Identity matrix:
For any square matrix 𝐴 = [𝑎𝑖𝑗 ], 𝑎𝑖𝑗 = 0, when 𝑖 ≠ 𝑗, and 𝑎𝑖𝑗 = 1, for 𝑖 = 𝑗 is called
an identity matrix. Unit matrix of order 𝑛 is denoted by 𝐼
1 0 0 
Example:- I = 0 1 0 
 
0 0 1 

Singleton matrix:
A matrix having only one element is called as a singleton matrix.
Example:- 𝐴 = [6]

2026-27 MATHEMATICS CET MATERIAL Page 2 of 16

Zero matrix:
A matrix in which each element is zero is called zero matrix or null matrix.
It is denoted by O
0 0 0
Example:O = [ ]
0 0 0
Triangular Matrix:
A Square Matrix [aij] is said to be triangular matrix if each elementabove or below the
principal diagonal is zero it is of two types-
(a) Upper Triangular Matrix :A Square Matrix [aij] is called the upper triangular Matrix,
if aij= 0 when i> j.
(b) Lower Triangular Matrix : A Square Matrix [aij] is called the lower Triangular Matrix,
if aij = 0 when i< j
n (n  1)
Note: 1. Minimum number of zero in a triangular matrix is given by where n isorder
2

of Matrix.
2. A diagonal matrix is both upper and lower triangular matrix.
Properties of Matrices Addition: If 𝐴 , 𝐵 𝑎𝑛𝑑 𝐶 are any matrices of order 𝑚 × 𝑛 then
1) Commutative Law : 𝐴 + 𝐵 = 𝐵 + 𝐴
2) Associative Law: (𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶)
3) 𝐴 + 𝑂 = 𝑂 + 𝐴 = 𝐴 , Where 𝑂 is the zero matrix of order 𝑚 × 𝑛
4) 𝐴 + [−𝐴] = [−𝐴] + 𝐴 = 𝑂, Where 𝑂 is the zero matrix of order 𝑚 × 𝑛
5) 𝑘[𝐴 + 𝐵] = 𝑘𝐴 + 𝑘𝐵, where 𝑘 is any scalar
Note: For addition and subtraction of matrices orders of matrices must be equal.
Properties of Matrix Multiplication:
If 𝐴 , 𝐵 𝑎𝑛𝑑 𝐶 are any matrices such that there multiplication exist then
1) Commutative Law: 𝐴𝐵 is need not be equal to 𝐵𝐴 ⟹ Matrix Multiplication is not
commutative.
2) Associative Law: (𝐴𝐵)𝐶 = 𝐴(𝐵𝐶) ⟹ Matrix multiplication is associative.
3) 𝐴𝐼 = 𝐼𝐴 = 𝐴 , Where 𝐼 is an identity matrix
4) 𝐴𝐵 = 𝑂 ⇏ 𝐴 = 𝑂 𝑜𝑟 𝐵 = 𝑂 , where 𝑂 is the zero matrix.
5) Distributive law : 𝐴[𝐵 + 𝐶] = 𝐴𝐵 + 𝐴𝐶
Note: 1. For multiplication of two matrices columns of the first matrix should be equal to
rows of the second matrix.
2. If the matrix 𝐴𝑚×𝑛 𝑎𝑛𝑑 𝐵𝑛×𝑝 , 𝑡ℎ𝑒𝑛 𝐴𝐵𝑚×𝑝
3. If 𝐴𝐵 is defined, then 𝐵𝐴 need not be defined.

2026-27 MATHEMATICS CET MATERIAL Page 3 of 16

Transpose of a Matrix:
If 𝐴 = [𝑎𝑖𝑗 ] is a matrix of order 𝑚 × 𝑛 then the matrix obtained from interchanging the
rows into columns of a matrix 𝐴 is called transpose of matrix.
If 𝐴 is the matrix then transpose of A is denoted by 𝐴′ or 𝐴𝑇
1 2
1 3 −1
Example: If 𝐴 = [ 3 1] then 𝐴′ = [ ]
2 1 2 2×3
−1 2 𝟑×𝟐
Note: If 𝐴 is a matrix of order 𝑚 × 𝑛 then 𝐴′ is the matrix of order 𝑛 × 𝑚
Properties:If 𝐴 and 𝐵 are any matrices such that there multiplication and addition exist then
1) (𝐴′ )′ = 𝐴 2) (𝐴 ± 𝐵)′ = 𝐴′ ± 𝐵 ′
3) (𝐴𝐵)′ = 𝐵 ′ 𝐴′ 4) (𝑘𝐴)′ = 𝑘𝐴′
Symmetric Matrix:-A matrix 𝐴 is said to be symmetric if 𝐴 = 𝐴′
2 3 1 2 3 1
Example: 𝐴 = [3 1 −2 ] ⟹ 𝐴′
= [ 3 1 −2] = 𝐴
1 −2 4 1 −2 4
⟹ 𝐴 = 𝐴′ ∴ A is symmetric matrix
Skew-Symmetric Matrix:- A matrix 𝐴 is said to be skew-symmetric if 𝐴 = −𝐴′
0 3 2 0 −3 −2 0 3 2
Example: 𝐴 = [−3 0 −1] ⟹ 𝐴′
= [ 3 0 1 ] = − [ −3 0 −1] = −𝐴
−2 1 0 2 −1 0 −2 1 0
⟹ 𝐴′ = −𝐴 ⟹ 𝐴 = −𝐴′ ∴ 𝐴 is skew-symmetric matrix
Note : (1)Principal diagonal elements of skew symmetric matrix is always zero.
(2) Trace of a skew symmetric matrix is always zero.
(3)Every square matrix 𝐴 can uniquely be expressed as sum of a symmetric and

skew symmetric matrix i.e. A =  (A  A T ) +  (A  A T )
1 1
2  2 

(4) If A is a square matrix, then 𝐴 + 𝐴′ , 𝐴𝐴′ , 𝐴′ 𝐴 are symmetric matrices while
𝐴 – 𝐴′ is skew symmetric Matrix.
(5) If 𝐴 is a symmetric Matrix, then – 𝐴 , 𝑘𝐴, 𝐴′ , 𝐴𝑛, 𝐴−1 , 𝐵 ′ 𝐴𝐵 are also symmetric
matrices where 𝑛 ∈ 𝑁, 𝑘 ∈ 𝑅 and 𝐵 is a square matrix of order that of 𝐴𝐴.
(6) If 𝐴 , 𝐵 are two symmetric matrices, then 𝐴 ± 𝐵, 𝐴𝐵 + 𝐵𝐴
are also symmetric matrices. 𝐴𝐵 − 𝐵𝐴 is a skew symmetric matrix, 𝐴𝐵is a
symmetric matrix when 𝐴𝐵 = 𝐵𝐴.
(7) If 𝐴 , 𝐵are two skew-symmetric matrices, then 𝐴 ± 𝐵, 𝐴𝐵 − 𝐵𝐴
are skew symmetric matrices, 𝐴𝐵 + 𝐵𝐴 is a symmetric matrix.

2026-27 MATHEMATICS CET MATERIAL Page 4 of 16

Invertible Matrix:
A square matrix 𝐴 of order 𝑛 is said to be invertible if there exist amatrix 𝐵 of
Order 𝑛 such that 𝐴𝐵 = 𝐵𝐴 = 𝐼 where 𝐼 is an identity matrix.
Properties of Inverse Matrix:
Let 𝐴 and 𝐵 are two invertible matrices of the same order, then
(i) (𝐴′ )−1 = (𝐴−1 )′
(ii) (𝐴𝐵)−1 = 𝐵 −1 𝐴−1
(iii) (𝐴𝑘 )−1 = (𝐴−1 )𝑘, 𝑘 ∈ 𝑁
(iv) 𝑎𝑑𝑗 (𝐴−1 ) = (𝑎𝑑𝑗 𝐴)−1
(v) (𝐴−1 )−1 = 𝐴
(vi) 𝐴 is symmetric matrix ⇒ 𝐴−1 is symmetric matrix.
(vii) 𝐴 is scalar matrix ⇒ 𝐴−1 is scalar matrix.
(viii) 𝐴 is diagonal matrix ⇒ 𝐴−1is diagonal matrix.
Note: 1) If 𝐴𝐵 = 𝐵𝐴 = 𝐼 then 𝐵 is called inverse of 𝐴 and it is denoted by 𝐴−1 .
𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼
**************************************
CLASSROOM CHALLENGE
1. Total number of elements in a matrix of order 𝟑 × 𝟏 is [E]
1) 1 2) 3 3) 9 4) 6
2. If 𝑨 is a matrix of order 𝟑 × 𝟒, then each row of 𝑨 has [E]
1) 3 elements 2) 4 elements 3) 12 elements 4) 7 elements
3. If A and B are two matrices of the order 𝟑 × 𝟒 and 𝟒 × 𝟐 respectively,
then the order of matrix 𝑨𝑩 is [A]
1) 3 × 4 2) 4 × 3 3) 2 × 3 4) 3 × 2
4. If 𝑨, 𝑩, 𝑪 are matrices of order𝟏 × 𝟑, 𝟑 × 𝟑 and 𝟑 × 𝟏 respectively,
the order of 𝑨𝑩𝑪 will be [E]
1) 3 × 3 2) 1 × 3 3) 1 × 1 4) 3 × 1
5. 𝑨 and 𝑩 are two matrices such that the order of 𝑨 is 𝟑 × 𝟒, if 𝑨𝑩 and 𝑩𝑨 are both
defined, then [A]
1) Order of 𝐵 is 3 × 4 2) Order of 𝐵𝐴 is 4 × 4
3) Order of 𝐵𝐴 is 3 × 3 4) 𝐵𝐴 is not defined
6. If A and B are two matrices such that AB is an identity matrix and the order of
matrix 𝑩 is 𝟑 × 𝟒 , then the order of 𝑨 is [E][KCET-25]
1) 3 × 4 2) 3 × 3 3) 4 × 3 4) 4 × 4

2026-27 MATHEMATICS CET MATERIAL Page 5 of 16

7. If 𝐀 is a matrix of order 3×4 and B is a matrix, such that 𝑨′ 𝐁 𝐚𝐧𝐝 𝐁𝑨′
are both defined, then the order of the matrix B is [A]
1) 4×3 2) 3×4 3) 3×3 4) 4×4
8. If 𝑨 and 𝑩 are two matrices such that 𝑨 + 𝑩 and 𝑨𝑩 are both defined, then [A]
1) 𝐴and𝐵 are two matrices not necessarily of same order
2) 𝐴and𝐵 are square matrices of same order
3) Number of columns of 𝐴 = number of rows of 𝐵
4) Number of rows of 𝐴 = number of columns of 𝐵
−𝟏 𝟓
−𝟏 𝟎 𝟐
9. If 𝑨 = [ ] 𝒂𝒏𝒅 𝑩 = [ 𝟐 𝟕 ] , then [E]
𝟑 𝟏 𝟐
𝟑 𝟏𝟎
1) AB and BA both exist 2) AB exists but not BA
3) BA exists but not AB 4) both AB and BA do not exist
10. The number of different orders of a matrix having 24 elements is [E]
1) 8 2) 1 3) 4 4) 6
11. The number of matrices of order 𝟑 × 𝟑 with each entry being a square root of unity
[E]
1) 81 2) 39 3) 256 4) 512
12. The number of 𝟐 × 𝟐matrices that can be formed by using 1, 2, 3, 4
when repetitions are allowed is [A]
1) 24 2) 12 3) 6 4) 256
13. A row matrix has only [E][KCET-26]
1) one element 2) one row with one or more columns
3) one column with one or more rows 4) one row and one column
14. For any square matrix 𝑨 = [𝒂𝒊𝒋 ], 𝒂𝒊𝒋 = 𝟎, when 𝒊 ≠ 𝒋, and 𝒂𝒊𝒋 = 𝒌,
where𝒌 is a constant for 𝒊 = 𝒋 is called [E]
1) Unit matrix 2) scalar matrix
3) diagonal matrix 4) row matrix
15. For any square matrix 𝑨 = [𝒂𝒊𝒋 ], 𝒂𝒊𝒋 = 𝒂𝒋𝒊 , then 𝑨 is [E]
1) Unit matrix 2) scalar matrix
3) diagonal matrix 4) symmetric matrix
16. If 𝑨 is skew symmetric matrix, then 𝑨𝟐𝟎𝟐𝟏 is [A][KCET-22]
1) Row Matrix 2) Symmetric Matrix
3) ColumnMatrix 4) Skew Symmetric Matrix

2026-27 MATHEMATICS CET MATERIAL Page 6 of 16

𝟐 𝒙+𝟑
17. If 𝑨 = [ ]is a symmetric matrix, then the value of 𝒙 is [E]
𝟐𝒙 + 𝟏 𝟑
1) 3 2) 5 3) 2 4) 4
18. If 𝑷 𝐚𝐧𝐝 𝑸 are symmetric matrices of the same order then 𝑷𝑸 − 𝑸𝑷 is [E][KCET-19]
1) Identity Matrix 2) Symmetric Matrix
3) ZeroMatrix 4) Skew Symmetric Matrix
19. If A is a skew symmetric matrix of order n and C is a column matrix of order 𝒏 × 𝟏,
then 𝑪′ 𝑨𝑪 is [D]
1) An identity matrix of order n 2) An identity matrix of order 1
3) An zero matrix of order n 4) An zero matrix of order 1
20. If A is a square matrix, then 𝑨 + 𝑨′ is a [E]
1) symmetric matrix 2) null matrix
3) unit matrix 4) skew symmetric matrix
21. Which of the following statements is not correct? [E][KCET-25]
1) A row matrix has only one row
2) A diagonal matrix has all diagonal elements equal to zero
3) A symmetric matrix A is a square matrix satisfying 𝐴′ = 𝐴
4) A skew symmetric matrix has all diagonal elements equal to zero
22. In the following, upper triangular matrix is- [A]
1 0 0 5 4 2 2 1
0 2 3
1) [0 2 0] 2) [0 0 3] 3) [ ] 4) [0 3]
0 0 4
3 0 3 0 0 1 0 0
3 0 0
23. Statement 1: Matrix [0 −2 0] is a scalar matrix. [E]
2 0 4
Statement 2: Every scalar matrix is a diagonal matrix.
Choose the correct answer from the option given below.
1) Statement 1 is false and Statement 2 is true.
2) Statement 1 is true and Statement 2 is false.
3) Statement 1 is false and Statement 2 is false.
4) Statement 1 is true and Statement 2 is true.
24. Minimum number of zeros in a triangular matrix of order 𝟑 × 𝟑 is [A]
1) 9 2) 6 3) 3 4) 0

2026-27 MATHEMATICS CET MATERIAL Page 7 of 16

𝒊
25. For 𝟐 × 𝟐 matrix, 𝑨 = [𝒂𝒊𝒋 ] whose elements are given by 𝒂𝒊𝒋 = 𝒋 then 𝑨 is equal to

[E]
1
2 3 1 1
1
1 2
1) [ 1 9] 2) [ 2 1] 3) [ 2] 4) [ ]
2 2 2 2 1 2 1
2

𝟏 𝟑 𝒚 𝟎 𝟓 𝟔
26. If 𝟐 [ ]+[ ]=[ ], then the value of 𝒙 𝒂𝒏𝒅 𝒚 are [E][KCET-17]
𝟎 𝒙 𝟏 𝟐 𝟏 𝟖
1) 𝑥 = −3, 𝑦 = −3 2) 𝑥 = −3, 𝑦 = 3 3) 𝑥 = 3, 𝑦 = −3 4) 𝑥 = 3, 𝑦 = 3
𝐜𝐨𝐬 𝟐𝜽 −𝐬𝐢𝐧 𝟐𝜽
27. If 𝑨 = [ ] 𝒂𝒏𝒅 𝑨+𝑨𝑻 = 𝑰 , then the value of 𝜽 is equal to[A][KCET-16]
𝐬𝐢𝐧 𝟐𝜽 𝐜𝐨𝐬 𝟐𝜽
𝜋 𝜋 3𝜋
1) 𝜋 2) 3 3) 6 4) 2

−𝟏 𝟏𝟖
𝟕 −𝟏𝟎 𝟏𝟕
28. If 𝟑𝑨 + 𝟒𝑩′ = [ ] 𝒂𝒏𝒅 𝟐𝑩 − 𝟑𝑨′ = [ 𝟒 𝟎 ]then 𝑩 = [A][KCET-19]
𝟎 𝟔 𝟑𝟏
−𝟓 −𝟕
1 3 1 −3 −1 −18 1 3
1) [−1 1] 2) [−1 1 ] 3) [ 4 −16] 4) [−1 1 ]
2 4 2 4 −5 −7 2 −4
𝟏 −𝟓 𝟕
29. If 𝑨 = [ 𝟎 𝟕 𝟗], then the value of tr (𝑨) is- [A]
𝟏𝟏 𝟖 𝟗
1) 25 2) 17 3) 3 4) 12
30. If 𝑨 is a skew symmetric matrix, then 𝒕𝒓(𝑨) = [A]
1) 0 2) 1 3) 2 4) −1
𝟐 −𝟐 𝟒
31. If 𝑩 = [−𝟏 𝟑 𝟒 ] = 𝑷 + 𝑸, where 𝑷 is a symmetric matrix and 𝑸 is a skew
𝟏 −𝟐 −𝟑
symmetric matrix, then 𝑷 = [A]
3 5 1 5
2 −2 0 −2 −2
2 −1 1 2 2 −2 1
3 1
1) [ 1 3 −4] 2) − 2 3 1 3) 2
0 3 4) [−2 3 4]
−1 4 −3 5 5 1 4 −3
[ 2 1 −3] [2 −3 0]

32. Which one of the following is not true [E]
1) matrix addition is commutative
2) matrix multiplication is commutative
3) matrix addition is associative
4) matrix multiplication is associative
33. If 𝑨 and 𝑩 are symmetric matrices of same order, then 𝑨𝑩 is symmetric if and
only if [E]
1) 𝐴𝐵 = 𝐼 2) 𝐴𝐵 = 𝑂 3) 𝐴𝐵 = 𝐵𝐴 4) 𝐴𝐵 = −𝐵𝐴

2026-27 MATHEMATICS CET MATERIAL Page 8 of 16

𝟐 𝟏
𝟏−𝟐 𝟏
34. If 𝑨 = [ ] 𝒂𝒏𝒅 𝑩 = [𝟑 𝟐], then (𝑨𝑩)′ is equal to [E][KCET-21]
𝟐𝟏 𝟑
𝟏 𝟏
−3 10 −3 −2 −3 7 −3 7
1) [ ] 2)[ ] 3) [ ] 4) [ ]
−2 7 10 7 10 2 10 −2
𝟐
35. If 𝑨 = [𝟏 𝟐 −𝟏] 𝒂𝒏𝒅 𝑩 = [𝟎], then 𝑨𝑩 is equal to [E]
𝟐
2 0 −2 2
1) [2 0 −2] 2) [0] 3) [2 0 −2] 4) [ 0 ]
2 0 −2 −2
𝟎 𝟎 𝟏
36. If 𝑨 = [𝟎 𝟏 𝟎] , then 𝑨𝟒 is equal to [D][KCET-20]
𝟏 𝟎 𝟎
1) 4𝐴 2) 𝐴 3) 2𝐴 4) I
𝟏 −𝟏 𝟏
37. If 𝑨 = [𝟏 −𝟏 𝟏] then, 𝑨𝟓 − 𝑨𝟒 − 𝑨𝟑 + 𝑨𝟐 = [D]
𝟏 −𝟏 𝟏
1) 2𝐴 2) 3𝐴 3) 4𝐴 4) 0
38. If A is a square matrix such that 𝑨𝟐 = 𝑨, then (𝑰 + 𝑨)𝟐 = [A]
1) 𝐼 + 𝐴 2) 3𝐴 + 𝐼 3) 3𝐼 + 𝐴 4) 𝐼 − 3𝐴
39. If A is a square matrix such that 𝑨𝟐 = 𝑨, then (𝑰 − 𝑨)𝟑 = [A][KCET-25]
1) 𝐼 − 𝐴 2) 𝐴 − 𝐼 3) 𝐼 + 𝐴 4) – 𝐼 − 𝐴
40. If 𝑨 is square matrix such that 𝑨𝟐 = 𝑰, then (𝑨 − 𝑰)𝟑 + (𝑨 + 𝑰)𝟑 − 𝟕𝑨 is equal to [D]
1) 𝐴 2) 𝐼 − 𝐴 3) 𝐼 + 𝐴 4) 3𝐴
41. If A is a square matrix such that 𝑨𝟐 = 𝑨 then (𝑰 + 𝑨)𝟒 = [D]
1) 𝐼 + 15𝐴 2) 𝐼 + 7𝐴 3) 𝐼 + 8𝐴 4) 𝐼 + 11𝐴
𝐜𝐨𝐬 𝜶 𝐬𝐢𝐧 𝜶
42. If 𝑨 = [ ] then 𝑨𝑨′ = [A] [KCET-18]
−𝐬𝐢𝐧 𝜶 𝐜𝐨𝐬 𝜶
1) 𝐴 2) 𝐴′ 3) 𝑂 4) I
43. If the matrix AB is a zero matrix, then [A]
1) 𝐴 = 0or𝐵 = 0 2) It is not necessary that either 𝐴 = 0 or 𝐵 = 0
3) 𝐴 = 0 and 𝐵 = 0 4) All the above statements are wrong
𝜶 𝟎 𝟏 𝟎
44. If 𝑨 = [ ] and 𝑩 = [ ] , then the value of 𝜶 when 𝑨𝟐 = 𝑩 [D]
𝟏 𝟏 𝟐 𝟏
1) 5 2) 1 3) −1 4) 11
𝜶 𝜷
45. If 𝑨 = [ ] is such that 𝑨𝟐 = 𝑰, then [A]
𝜸 −𝜶
1) 1 + 𝛼 2 + 𝛽𝛾 = 0 2) 1 − 𝛼 2 + 𝛽𝛾 = 0
3) 1 − 𝛼 2 − 𝛽𝛾 = 0 4) 1 + 𝛼 2 − 𝛽𝛾 = 0

2026-27 MATHEMATICS CET MATERIAL Page 9 of 16

𝟏 𝟏
46. If a matrix 𝑨 = [ ] satisfies 𝑨𝟔 = 𝒌𝑨′ , then the value of k is [D][KCET-25]
𝟏 𝟏
1
1) 32 2) 1 3) 32 4) 6
𝟑 𝟏
47. If 𝑨 = [ ] , then 𝑨𝟐 − 𝟓𝑨 = [A][KCET-16]
−𝟏 𝟐
1) I 2) −I 3) 7I 4) −7I
𝟑 −𝟐
48. If 𝑨 + 𝑰 = [ ] , then (𝑨 + 𝑰)(𝑨 − 𝑰) is equal to [D]
𝟒 𝟏
−5 −4 −5 4 5 4 −5 −4
1) [ ] 2) [ ] 3) [ ] 4) [ ]
8 −9 −8 9 8 9 −8 −9
𝟏⁄ 𝟑⁄
49. Let 𝑨 = [ 𝟐 𝟒 ]then the sum of all the elements of 𝑨𝟏𝟎𝟎 is [A]
𝟏 −𝟏 ⁄𝟐
3 1
1) 0 2) 2 3) 1 4 4) 1 16
𝟏 𝟓
−𝟒
𝟎 𝟏 𝟑 𝟐 𝟐
𝟏 𝟑
50. If 𝑨 = [𝟏 𝟐 𝒙] 𝒂𝒏𝒅 𝑨−𝟏 = − 𝟐 𝟑 − 𝟐 then the value of 𝒙𝒚 is [A]
𝟐 𝟑 𝟏 𝟏 𝟏
[ 𝟐 𝒚 𝟐 ]

1) 1 2) – 1 3) 0 4) 2
𝟎 𝟐𝒚 𝒛
51. If 𝑨 = [𝒙 𝒚 −𝒛]and 𝑨′ 𝑨 = 𝑰, then 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐 = [D]
𝒙 −𝒚 𝒛
1) 3 2) 0 3) 2 4) 1
𝟏 𝟏
52. If 𝑨 = [ ] then 𝑨𝟏𝟎 is equal to [A][KCET-24]
𝟏 𝟏
1) 28 𝐴 2) 29 𝐴 3) 210 𝐴 4) 211 𝐴
53. If 𝒎[−𝟑 𝟒] + 𝒏[𝟒 −𝟑] = [𝟏𝟎 −𝟏𝟏] , then 𝟑𝒎 + 𝟕𝒏 = [D]
1) 10 2) 5 3) 1 4) 3
𝟏 𝟏 𝒙 𝟐
54. If [ ] [𝒚] = [ ] then the value of 𝒙 𝒂𝒏𝒅 𝒚 respectively are [E][KCET-18]
−𝟏 𝟏 𝟒
1) −3, −1 2) 1, 3 3) 3, 1 4) −1, 3
𝟎 𝟏 𝒙 𝟑
55. If [ ] [𝒚] = [ ] then the value of 𝒙 𝒂𝒏𝒅 𝒚 respectively are [E]
−𝟏 𝟎 𝟓
1) −5, 3 2) 5, 3 3) −5, −3 4) 5, −3
56. If 𝑨 and 𝑩 are square matrices, then (𝑨 + 𝑩)𝟐 is equal to [A]
1) 𝐴2 + 2𝐴𝐵 + 𝐵 2 2) 𝐴2 + 𝐵 2 3) 𝐴2 + 2𝐵𝐴 + 𝐵 2 4) 𝐴2 + 𝐴𝐵 + 𝐵𝐴 + 𝐵 2
57. If 𝑨 and 𝑩 are two matrices such that 𝑨𝑩 = 𝑩 and 𝑩𝑨 = 𝑨, then 𝑨𝟐 + 𝑩𝟐 =
[D][KCET-23]
1) 2 𝐵𝐴 2) 𝐴 + 𝐵 3) 2 𝐴𝐵 4) 𝐴𝐵

2026-27 MATHEMATICS CET MATERIAL Page 10 of 16

𝟏 𝟐 𝟐
58. If 𝑨 = [𝟐 𝟏 𝟐] , then 𝑨𝟐 − 𝟒𝑨 is equal to [A]
𝟐 𝟐 𝟏
1) 2𝐼3 2) 3𝐼3 3) 4𝐼3 4) 5𝐼3
59. If for a matrix 𝑨, 𝑨𝟑 = 𝑰 then 𝑨−𝟏 is equal to [A]
1) 𝐴2 2) 𝐴3 3) 𝐴 4) 𝐼
60. If 𝑨 is a matrix of order 𝟑 × 𝟑 , then (𝑨𝟐 )−𝟏 = [E][KCET-22]
1) −(𝐴2 )2 2) 𝐴2 3) (𝐴−1 )2 4) −(𝐴)−2

CLASSROOM CHALLENGE
1 2 3 4 5 6 7 8 9 10
2 2 4 3 2 3 2 2 1 1
11 12 13 14 15 16 17 18 19 20
4 4 2 2 4 4 3 4 4 1
21 22 23 24 25 26 27 28 29 30
2 2 1 3 3 4 3 1 2 1
31 32 33 34 35 36 37 38 39 40
2 2 3 1 2 2 4 2 1 1
41 42 43 44 45 46 47 48 49 50
1 3 2 2 3 1 4 1 2 2
51 52 53 54 55 56 57 58 59 60
1 2 3 4 1 4 2 4 1 3

ADDITIONAL PROBLEMS
1. If 𝑨 is a matrix of order 𝒎 × 𝒏 and 𝑩 is a matrix such that 𝑨𝑩′ and 𝑩′𝑨 are both
defined, then order of the matrix B is [A][KCET-16]
1) 𝑚 × 𝑚 2) 𝑛 × 𝑛 3) 𝑛 × 𝑚 4) 𝑚 × 𝑛
2. If 𝑨 is a matrix of order 𝟑 × 𝟐 and 𝑩 is a matrix of order 𝟒 × 𝟐 then the order of
𝑨𝑩′ is [E]
1) 3 × 2 2) 2 × 2 3) 3 × 4 4) 4 × 3
3. If 𝑨 is a matrix of order 𝟑 × 𝟐 and 𝑩 is a matrix of order 𝟒 × 𝟐
then the order of ABis [E]
1) 3 × 2 2) 2 × 2 3) 4 × 3 4) not defined
4. If 𝑨 is a matrix of order 𝟑 × 𝟓 and 𝑩 is a matrix of order 𝟐 × 𝟑
then the order of 𝑩𝑨 is [E]
1) 3 × 2 2) 2 × 5 3) 5 × 2 4) 2 × 3
5. If the order of the matrix A is 𝟐 × 𝟑, the order of the matrix B is 𝟑 × 𝟐 and the order
of the matrix C is 𝟑 × 𝟑, then which of the following is not defined [A]
1) 𝐶𝐵 + 𝐴′ 2) 𝐵𝐴𝐶 3) 𝐶(𝐴 + 𝐵 ′ )′ 4) 𝐶(𝐴 + 𝐵 ′ )
2026-27 MATHEMATICS CET MATERIAL Page 11 of 16

6. If the order of 𝑨 is 𝟒 × 𝟑 and the order of 𝑩 is 𝟒 × 𝟓, then the order of (𝑨𝑻 𝑩)𝑻 is [E]
1) 5 × 3 2) 3 × 4 3) 4 × 3 4) 3 × 5
7. If A and B are two matrices of the order 𝟑 × 𝒎 and 𝟑 × 𝒏, respectively and 𝒎 = 𝒏,
then the order of matrix (𝟓𝑨 − 𝟐𝑩) is [A]
1) 𝑚 × 3 2) 3 × 3 3) 𝑚 × 𝑛 4) 3 × 𝑛
8. The number of different orders of a matrix having 5 elements is [E]
1) 3 2) 1 3) 2 4) 5
9. The number of matrices of order 𝟑 × 𝟑 with each entry 0 and 1 [E]
1) 9 2) 27 3) 256 4) 512
10. If a matrix 𝑨 is both symmetric and skew symmetric, then 𝑨 is [E][KCET-17]
1) Diagonal matrix 2) Zero matrix 3) Scalar matrix 4) Square matrix
11. If 𝑨 is skew symmetric matrix, then 𝑨𝟐𝟎 is [A]
1) Row Matrix 2) Symmetric Matrix
3) ColumnMatrix 4) Skew Symmetric Matrix
12. If A is a skew symmetric matrix and n a positive even integer, then 𝑨𝒏 is a [A]
1) Symmetric matrix 2) Skew symmetric matrix
3) Diagonal matrix 4) Scalar matrix
13. If A is a square matrix, then 𝑨 − 𝑨′ is a [E]
1) unit matrix 2) null matrix
3) symmetric matrix 4) skew symmetric matrix
14. If 𝑨 and 𝑩 are square matrices of same order and 𝑩 is a skew symmetric matrix,
then 𝑨′ 𝑩𝑨 is [A][KCET-20]
1) Null Matrix 2) Symmetric Matrix
3) DiagonalMatrix 4) Skew Symmetric Matrix
𝒙−𝟐 𝒚+𝟒
15. If 𝑨 = [ ] is a scalar matrix, then the value of 𝒙 𝒂𝒏𝒅 𝒚 are [E]
𝟎 𝟕
1) 𝑥 = 9, 𝑦 = −4 2) 𝑥 = −4, 𝑦 = 9 3) 𝑥 = 2, 𝑦 = 4 4) 𝑥 = 2, 𝑦 = −4
16. In the following, lower triangular matrix is- [A]
5 4 2 1 0 0 2 1
0 2 3
1) [ ] 2) [0 0 3] 3) [0 2 0] 4) [0 3]
0 0 4
0 0 1 3 0 3 0 0
17. If the trace of a matrix 𝑨 is 7, then the trace of a matrix 𝟑𝑨 is [A]
1) 3 2) 10 3) 2 4) 21
𝟐 𝟒 𝟎 𝟔 −𝟐 𝟒
18. Let 𝑨 + 𝟐𝑩 = [ 𝟔 −𝟑 𝟑] and 𝑨 − 𝟐𝑩 = [𝟔 𝟏 𝟓] then 𝒕𝒓(𝑨) = [A]
−𝟓 𝟑 𝟓 𝟔 𝟑 𝟑
1) 17 2) 5 3) 29 4) 7
2026-27 MATHEMATICS CET MATERIAL Page 12 of 16

𝟏 𝟐 𝟑 𝟖
19. If 𝟐𝑿 + [ ]=[ ] , then 𝑿 is equal to [A]
𝟑 𝟒 𝟕 𝟐
1 3 1 −3 2 6 2 −6
1) [ ] 2) [ ] 3) [ ] 4) [ ]
2 −1 2 −1 4 −2 4 −2
𝟏, 𝒊𝒇 𝒊 ≠ 𝒋
20. If matrix 𝑨 = [𝒂𝒊𝒋 ] , where 𝒂𝒊𝒋 = { , then 𝑨𝟐 is equal to [A]
𝟎, 𝒊𝒇 𝒊 = 𝒋
1) 𝐼 2) 𝐴 3) 𝑂 4) 𝐴2
21. If A is a square matrix such that 𝑨𝟐 = 𝑨 then (𝑰 + 𝑨)𝟑 = [A][KCET-24]
1) 7A − I 2) 7A 3) 7A + I 4) I − 7A
22. If 𝑨 is square matrix such that 𝑨𝟐 = 𝑨, then (𝑰 + 𝑨)𝟑 − 𝟕𝑨 is equal to [A]
1) 𝐼 2) 𝐼 − 𝐴 3) 𝐴 4) 3𝐴
23. If A is a square matrix such that 𝑨𝟐 = 𝑨, then (𝑰 − 𝑨)𝟐 = [A]
1) 𝐼 − 𝐴 2) 𝐴 − 𝐼 3) 𝐼 + 𝐴 4) – 𝐼 − 𝐴
𝐜𝐨𝐬 𝜽 𝐬𝐢𝐧 𝜽
24. If 𝑨 = [ ] then 𝑨𝟐 = [A]
𝐬𝐢𝐧 𝜽 −𝐜𝐨𝐬 𝜽
1) 𝐴 2) 𝐴′ 3) 𝑂 4) I
𝒂 𝒃 𝜶 𝜷
25. If 𝑨 = [ ] and 𝑨𝟐 = [ ] , then [D]
𝒃 𝒂 𝜷 𝜶
1) = 2𝑎𝑏 , 𝛽 = 𝑎2 + 𝑏 2 2) 𝛼 = 𝑎2 + 𝑏 2 , 𝛽 = 𝑎𝑏
3) = 𝑎2 + 𝑏 2 , 𝛽 = 2𝑎𝑏 4) 𝛼 = 𝑎2 + 𝑏 2 , 𝛽 = 𝑎2 − 𝑏 2
𝟎 𝟏
26. If 𝑨 = [ ] , then 𝑨𝟐 = [A][KCET-15]
𝟏 𝟎
0 1 1 0 1 0 0 1
1) [ ] 2) [ ] 3) [ ] 4) [ ]
1 0 1 0 0 1 0 1
𝟎 𝒊
27. If 𝑨 = [ ] , then 𝑨𝟒𝟎 = [A]
−𝒊 𝟎
0 1 1 0 0 𝑖 0 −𝑖
1) [ ] 2) [ ] 3) [ ] 4) [ ]
1 0 0 1 −𝑖 0 𝑖 0
𝟐 −𝟐
28. If 𝑨 = [ ] then 𝑨𝒏 = 𝟐𝒌 𝑨, where 𝒌 = [D][KCET-18]
−𝟐 𝟐
1) 2(𝑛 − 1) 2) 𝑛 + 1 3) 2𝑛−1 4) 𝑛 − 1
𝟒 −𝟒 𝟖 𝟒
29. If [𝟏] 𝑨 = [−𝟏 𝟐 𝟏] then 𝑨 = [A]
𝟑 −𝟑 𝟔 𝟑
1) doesn’t exist 2) [–1 2 –1] 3) [–1 2 1] 4) [–1 –2 1]
𝟎 𝒂𝟒
30. If [ ] = 𝑰, then [D]
𝒃 𝟎
1) 𝑎 = 1 = 2𝑏 2) 𝑎 = 𝑏 3) 𝑎 = 𝑏 2 4) 𝑎𝑏 = 1

2026-27 MATHEMATICS CET MATERIAL Page 13 of 16

𝟎 𝟎 −𝟏
31. Let 𝑨 = [ 𝟎 −𝟏 𝟎 ] The only correct statement about the matrix A is [A]
−𝟏 𝟎 𝟎
1) 𝐴is a zero matrix 2) 𝐴 = (– 1) I, where I is a unit matrix
3) 𝐴−1 does not exist 4) 𝐴2 = I
𝟐 −𝟏 𝒏
32. If n is a natural number, then [ ] , is [D]
𝟑 −𝟐
1 0 1 0
1) [ ]ifn is even 2) [ ] if n is odd
0 1 0 1
−1 0 1 0
3) [ ]if𝑛 ∈ N 4) [ ] if 𝑛 ∈ N
0 1 0 1
𝟑 𝟏 𝟏𝟓
33. If 𝒙 [ ] + 𝒚 [ ] = [ ] then the value of 𝒙 𝒂𝒏𝒅 𝒚 are [E][KCET-23]
𝟐 −𝟏 𝟓
1) 𝑥 = −4, 𝑦 = 3 2) 𝑥 = 4, 𝑦 = 3 3) 𝑥 = 4, 𝑦 = −3 4) 𝑥 = −4, 𝑦 = −3
𝟎 𝟐 𝟎 𝟑𝒂
34. If 𝑨 = [ ]and 𝒌𝑨 = [ ] , then the values of a, b, k are respectively
𝟑 −𝟒 𝟐𝒃 𝟐𝟒
[D]
1) −6, −12, −18 2) −6, 4, 9 3) −6, −4, −9 4) −6, 12, 18

35. If 𝑨 = [ 𝒂𝒃𝟐 𝒃𝟐 ]and 𝑨𝒏 = 𝟎 , then the minimum value of ‘𝒏’ is [D]
−𝒂 −𝒂𝒃
1) 2 2) 3 3) 4 4) 5
36. If 𝑨𝑩 = 𝑨 and 𝑩𝑨 = 𝑩 then [A]
1) 𝐴 = 2𝐵 2) 𝐴2 = 𝐴 and 𝐵 2 = 𝐵
3) 2𝐴 = 𝐵 4) cannot be determined
𝒙 𝒙
𝒔𝒊𝒏−𝟏 𝝅𝒙 𝒕𝒂𝒏−𝟏 𝝅 −𝒄𝒐𝒔−𝟏 𝝅𝒙 𝒕𝒂𝒏−𝟏 𝝅 𝟏
37. If 𝑨 = [ 𝒙 ] and 𝑨 = [ 𝒙 ] , then (𝑨 − 𝑩) is[H]
𝒔𝒊𝒏−𝟏 𝝅 𝒕𝒂𝒏 −𝟏
𝝅𝒙 𝒔𝒊𝒏−𝟏 𝝅 −𝒄𝒐𝒕 −𝟏
𝝅𝒙 𝛑

1
1) 0 2) 2 𝐼 3) 2𝐼 4) 3𝐼

38. Consider the following statements [A]
i) If A and B are two square matrices of same order and commute, then
(𝐴 + 𝐵)(𝐴 − 𝐵) = 𝐴2 − 𝐵 2
ii) If A and B are two square matrices of same order, then (𝐴𝐵)𝑛 = 𝐴𝑛 𝐵 𝑛
iii) If A and B are two matrices such that AB=A and BA=B, then A and B
are idempotent
Which of these is/are correct?
1) only (i) 2) (ii) and (iii) 3) (iii) and (i) 4) all of these
39. If 𝑨 and 𝑩 are square matrices, then (𝑨 − 𝑩)𝟐 is equal to [A]
1) 𝐴2 − 2𝐴𝐵 + 𝐵 2 2) 𝐴2 − 𝐵 2 3) 𝐴2 − 2𝐵𝐴 + 𝐵 2 4) 𝐴2 − 𝐴𝐵 − 𝐵𝐴 + 𝐵 2

2026-27 MATHEMATICS CET MATERIAL Page 14 of 16

40. The inverse of a symmetric matrix is [E]
1) Symmetric matrix 2) Skew symmetric matrix
3) Diagonal matrix 4) Identity Matrix

ADDITIONAL PROBLEMS
1 2 3 4 5 6 7 8 9 10
4 3 4 2 4 1 4 3 4 2
11 12 13 14 15 16 17 18 19 20
2 1 4 4 1 3 4 4 1 1
21 22 23 24 25 26 27 28 29 30
3 1 1 4 3 3 2 1 3 4
31 32 33 34 35 36 37 38 39 40
4 1 2 3 1 2 2 3 4 1

2026-27 MATHEMATICS CET MATERIAL Page 15 of 16

2026-27 MATHEMATICS CET MATERIAL Page 16 of 16