Karnataka 2nd PUC Mathematics Determinants 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: DETERMINANTS

Definition:
To every square matrix 𝐴 = [𝑎𝑖𝑗 ] of order n, we can associate a number called determinant
of a square matrix 𝐴. The determinant of the matrix A is denoted by |𝐴| 𝑜𝑟 ∆ 𝑜𝑟 𝑑𝑒𝑡 𝐴.
Note: Sign of the 2 × 2 determinant is as follows
+ −
| |
− +
Sign of the 3 × 3 determinant is as follows
+ − +
|− + −|
+ − +
Expansion of 𝟐 × 𝟐 determinant:
𝑎11
𝑎12
If 𝐴 = [𝑎 𝑎 ] is a square matrix of order 2 then
21 22
𝑎11 𝑎12
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝐴 = |𝐴| = |𝑎 | = 𝑎11 𝑎22 − 𝑎12 𝑎21
21 𝑎22

Expansion of 𝟑 × 𝟑 determinant:
𝑎1 𝑎2
𝑎3
If 𝐴 = [𝑏1 𝑏3 ] is a square matrix of order 3 then
𝑏2
𝑐1 𝑐2
𝑐3
𝑎1 𝑎2 𝑎3
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝐴 = |𝐴| = |𝑏1 𝑏2 𝑏3 |
𝑐1 𝑐2 𝑐3 3×3
𝑏 𝑏3 𝑏 𝑏3 𝑏 𝑏2
⟹ |𝐴| = 𝑎1 | 2 | − 𝑎2 | 1 | + 𝑎3 | 1 |
𝑐2 𝑐3 2×2 𝑐1 𝑐3 2×2 𝑐1 𝑐2 2×2
⟹ |𝐴| = 𝑎1 (𝑏2 𝑐3 − 𝑐2 𝑏3 ) − 𝑎2 (𝑏1 𝑐3 − 𝑐1 𝑏3 ) + 𝑎3 (𝑏1 𝑐2 − 𝑐1 𝑏2 )
Note: The determinant value of the triangular matrix is the product of the diagonal
elements.

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Properties of Determinants:
 The value of a determinant is unaltered if it’s rows and columns are interchanged
(𝑖. 𝑒. |𝐴| = |𝐴′ |)
 If two rows (or columns) of a determinant are interchanged then sign of the determinant
changes
 If in a determinant two rows (or columns) are identical then the value of the determinant is
zero.
 If the elements of any row (or column) are zero then the value of the determinant is zero.
 If the elements of any row(or column) are multiplied by k then the value of the determinant
is multiplied by k.
 If each element in any row (or column) of a determinant is a sum of two terms then the
determinant can be expressed as a sum of 2 determinants.
 If to the elements of any row (or column) of a determinant the same multiples of the
corresponding elements of the other rows (or columns) of the determinant are added then the
value of the determinant is unaltered
 The value of a determinant of upper triangular matrix or lower triangular matrix is equal to
product of the principal diagonal element.
Area of the triangle formed by the vertices (𝒙𝟏 , 𝒚𝟏 ), (𝒙𝟐 , 𝒚𝟐 ) & (𝒙𝟑 , 𝒚𝟑 ) is given by
𝒙𝟏 𝒚𝟏 𝟏
𝟏
∆= 𝟐 |𝒙𝟐 𝒚𝟐 𝟏|
𝒙𝟑 𝒚𝟑 𝟏
Equation of the line passing through the points (𝒙𝟏 , 𝒚𝟏 ) and (𝒙𝟐 , 𝒚𝟐 ) is given by
𝒙𝟏 𝒚𝟏 𝟏
|𝒙𝟐 𝒚𝟐 𝟏| = 𝟎
𝒙 𝒚 𝟏
Note: - 1) |𝑘𝐴| = 𝑘 𝑛 |𝐴| where 𝑘 is a constant and 𝑛 is the order of the matrix 𝐴
2) If 𝐴 and 𝐵 are the square matrices of same order then |𝐴𝐵| = |𝐴||𝐵|
3) The determinant of a skew symmetric matrix of odd order is 0
4) The determinant of a skew symmetric matrix of even order is a perfect square.
Minors, Cofactors and Adjoint:
Minor: The minor of an element 𝑎𝑖𝑗 is the value of the determinant obtained by deleting 𝑖 𝑡ℎ row
and 𝑗 𝑡ℎ column of the matrix. The minor of the element 𝑎𝑖𝑗 is denoted by 𝑀𝑖𝑗 .
Cofactor: The cofactor of an element 𝑎𝑖𝑗 of a matrix is denoted and defined by𝐴𝑖𝑗 = (−1)𝑖+𝑗 𝑀𝑖𝑗 ,
Where 𝑀𝑖𝑗 is minor of the element 𝑎𝑖𝑗

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1 2 3
Example: Let 𝐴 = (4 5 6) be any 3 × 3 matrices.
7 8 9
1 2
Now Minor of element 6 = 𝑀23 = | | = 8 − 14 = −6
7 8
Now cofactor of element 6 = 𝐴23 = (−1)2+3 𝑀23 = (−1)5 (−6) ⟹ 𝐴23 = (−1)(−6) = 6
Cofactor matrix:
The cofactor matrix of a square matrix 𝐴 is the matrix obtained by replacing the elements
of 𝐴 by its corresponding cofactors
Adjoint of the Matrix:
The adjoint of the matrix 𝐴 is the transpose of the cofactor matrix 𝐴 and it is denoted by 𝑎𝑑𝑗𝐴.
If the elements of any row (or column) are multiplied by there corresponding cofactors and
added, then there sum is the value of the determinant.
𝒂𝟏𝟏 𝑨𝟏𝟏 + 𝒂𝟏𝟐 𝑨𝟏𝟐 + 𝒂𝟏𝟑 𝑨𝟏𝟑 = |𝑨|
If the elements of a row (or column) are multiplied by the cofactor of the corresponding
elements of any other row (or column) and added, then their sum is zero.
𝒂𝟐𝟏 𝑨𝟏𝟏 + 𝒂𝟐𝟐 𝑨𝟏𝟐 + 𝒂𝟐𝟑 𝑨𝟏𝟑 = 𝟎
Note: 1) If 𝐴 is a square matrix of order 𝑛 and |𝐴| ≠ 0 then |𝑎𝑑𝑗 𝐴| = |𝐴|𝑛−1
2) For a square matrix, 𝐴(𝑎𝑑𝑗 𝐴) = (𝑎𝑑𝑗 𝐴)𝐴 = |𝐴|𝐼
𝐴
3) (𝑎𝑑𝑗 𝐴)−1 = 𝑎𝑑𝑗(𝐴−1 ) = |𝐴|

4) 𝑎𝑑𝑗 𝐴𝑇 = (𝑎𝑑𝑗 𝐴)𝑇
5) |𝐴−1 | = |𝐴|−1
6) 𝑎𝑑𝑗(𝑎𝑑𝑗 𝐴) = |𝐴|𝑛−2 𝐴
7) 𝑎𝑑𝑗(𝑘𝐴) = 𝑘 𝑛−1 𝑎𝑑𝑗 𝐴
Singular Matrix:
A square matrix 𝐴 is said to be singular if |𝐴| = 0
Non singular Matrix:
A square matrix 𝐴 is said to be non-singular if |𝐴| ≠ 0
Inverse of a matrix:
Let 𝐴 be a square matrix. If 𝐵 is matrix such that 𝐴𝐵 = 𝐵𝐴 = 𝐼,
then 𝐵 is called inverse of 𝐴 and it is denoted by 𝐴−1 .
𝟏
Inverse of the matrix exist if 𝑨 is non singular and 𝑨−𝟏 = |𝑨| (𝒂𝒅𝒋 𝑨).

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Consistency and Inconsistency of a System of Equation:
A system of equation is said to be consistent if it has a solution.
A system of equation is said to be inconsistent if it has no solution.
System of linear equations 𝒂𝟏 𝒙 + 𝒃𝟏 𝒚 = 𝒄𝟏 and 𝒂𝟐 𝒙 + 𝒃𝟐 𝒚 = 𝒄𝟐 has
𝑎1 𝑏
(i) Unique solution if, ≠ 𝑏1
𝑎 2 2

𝑎1 𝑏 𝑐
(ii) Infinitely many solutions if, = 𝑏1 = 𝑐1
𝑎2 2 2

𝑎1 𝑏 𝑐
(iii) No solution if, = 𝑏1 ≠ 𝑐1
𝑎2 2 2

System of linear equations
𝒂𝟏 𝒙 + 𝒃𝟏 𝒚 + 𝒄𝟏 𝒛 = 𝒅𝟏 , 𝒂𝟐 𝒙 + 𝒃𝟐 𝒚 + 𝒄𝟐 𝒛 = 𝒅𝟐 and 𝒂𝟑 𝒙 + 𝒃𝟑 𝒚 + 𝒄𝟑 𝒛 = 𝒅𝟑 has
(i) Unique solution if, ∆≠ 0
(ii) Infinitely many solutions if, ∆= 0, ∆1 = ∆2 = ∆3 = 0
(iii) No solution if, at least one of ∆1 , ∆2 , ∆3 ≠ 0
********************************************

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CLASSROOM CHALLENGE
1. Which of the following is correct [E]
1) Determinant is a matrix
2) Determinant is a number associated to a matrix
3) Determinant is a number associated to a square matrix
4) Determinant is a square matrix
𝟑 𝒙 𝟑 𝟐
2. If | |=| | then x is equal to [E] [KCET-17]
𝒙 𝟏 𝟒 𝟏
1) 2 2) ±8 3) 4 4) ±2√2
𝟎
3. Evaluate |𝐜𝐨𝐬 𝟏𝟓𝟎 𝐬𝐢𝐧 𝟏𝟓𝟎 | [A]
𝐬𝐢𝐧 𝟕𝟓 𝐜𝐨𝐬 𝟕𝟓𝟎
1) 1 2) 0 3) 2 4) 3
𝟒 + 𝒊 −𝟐𝒊
4. The value of | | is [A]
𝟑𝒊 𝟒−𝒊
1) 12 2) 17 3) 11 4) 20
𝟐 −𝟏 𝟐
5. The value of the determinant |𝟎 𝟑 𝟒| [E]
𝟎 𝟎 𝟓
1) 10 2) 30 3) 20 4) 40
𝟐 −𝟏 𝟐
6. If ∆= | 𝟐 𝒙 𝟒 |=𝟎 find the value of x [E]
−𝟏 𝟏 −𝟐
1) −2 2) 4 3) −4 4) 2
7. The value of an even order skew symmetric determinant is [E]
1) 0 2) perfect square 3) ±1 4) 2
8. If |A|=8 then the value of |𝑨𝑨′| = [E]
1
1) 1 2) 8 3) 64 4) 2
𝟑 𝟏 𝟐
9. If the value of the determinant |𝟒 𝟎 −𝟏| = 𝒌, then the value of the determinant
𝟓 −𝟐 −𝟓
𝟒 𝟎 −𝟏
|𝟑 𝟏 𝟐 | is [A]
𝟓 −𝟐 −𝟓
1) 𝑘 2) −𝑘 3) 𝑘 2 4) 2𝑘
𝟏 −𝟐 𝟓
10. There are two values of a which makes determinant ∆= |𝟐 𝒂 −𝟏| = 𝟖𝟔 then the
𝟎 𝟒 𝟐𝒂
sum of these numbers is [A][ KCET-22]
1) −4 2) 4 3) 9 4) 5

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𝐜𝐨𝐬(𝒙 + 𝒚) − 𝐬𝐢𝐧(𝒙 + 𝒚) 𝐜𝐨𝐬 𝟐𝒚
11. The determinant | 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚 | is independent of [D]
− 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚
1) both x and y 2) x but not of y 3) y but not of x 4) neither x nor y
𝒂+𝟏 𝟏 𝟏
12. If | 𝟏 𝟏 −𝟏| = 4, then the value of a is [A]
−𝟏 𝟏 𝟏
1) 1 2) –1 3) –2 4) 0
𝟏 𝟐 −𝟏
13. If 𝑨 = [𝟏 𝒙−𝟐 𝟏 ], is singular then the value of x is [A]
𝒙 𝟏 𝟏
1) 2 2) 3 3) 1 4) 0
𝟎 𝒙−𝒂 𝒙−𝒃
14. If 𝒇(𝒙) = |𝒙 + 𝒂 𝟎 𝒙 − 𝒄 | , then [H]
𝒙+𝒃 𝒙+𝒄 𝟎
1) 𝑓(𝑎) = 0 2) 𝑓(𝑏) = 0 3) 𝑓(0) = 0 4) 𝑓(1) = 0
𝒂 𝟔
15. If 𝑨 = [ ] is a singular matrix, then the value of 𝒂 is [E]
𝟔 𝟑
1) 3 2) −3 3) 12 4) 6
𝟐−𝒌 𝟐
16. If 𝑨 = [ ] is singular matrix , then the value of 𝟓𝒌 − 𝒌𝟐 is equal to
𝟏 𝟑−𝒌
[A] [KCET-23]
1) 6 2) 4 3) −6 4) −4
𝒌 𝟐
17. If 𝑨 = [ ] and |𝑨𝟑 | = 𝟏𝟐𝟓, then the value of 𝒌 is [A] [KCET-25]
𝟐 𝒌
1) ±2 2) ±3 3) −5 4) −4
18. If |𝑨𝑩| = 𝟏𝟔, |𝑨| = 𝟖 , where 𝑨 and 𝑩 are square matrices of same order, then |𝑩| is
[E]
1
1) 2 2) −2 3) 8 4) 8

19. If A is any square matrix of order 𝟑 × 𝟑, then |𝒌𝑨| is equal to [E] [KCET-17]
1) 𝑘|𝐴| 2) 𝑘 2 |𝐴| 3) 𝑘 3 |𝐴| 4) 3𝑘|𝐴|
𝟏 𝟎 𝟏
20. If A=[𝟎 𝟏 𝟐], then |𝟑𝐀| is equal to [A]
𝟎 𝟎 𝟒
1) 27 2) 4 3) 54 4) 108
21. If A is any square matrix of order 𝟑 × 𝟑 then |𝟑𝑨| is equal to [E] [KCET-16]
1
1) 27|𝐴| 2) 3|𝐴| 3) 9|𝐴| 4) 3 |𝐴|

22. If each row of a determinant of third order of value ∆ is multiplied by 3, then the
value of new determinant is [E]
1) ∆ 2) 27 ∆ 3) 21 ∆ 4) 54 ∆
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23. If A and B are matrices of order 3 and |𝑨| = 𝟓, |𝑩| = 𝟑 then |𝟑𝑨𝑩| is [E] [KCET-21]
1) 425 2) 405 3) 565 4) 585
24. If 𝑨 is a square matrix of order 𝟑 and |𝑨| = 𝟓, then the value of |𝟐𝑨′| is [E]
1) −10 2) 10 3) −40 4) 40
25. If the vertices of a triangle are (−𝟐, 𝟔), (𝟑, −𝟔) 𝒂𝒏𝒅 (𝟏, 𝟓) then the area of the triangle
is [E] [KCET-22]
1) 40 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠 2) 30 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠 3) 15.5 𝑠𝑞. 𝑢𝑛𝑖𝑡 4) 35 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠
26. If (𝒙𝟏 , 𝒚𝟏 ), (𝒙𝟐 , 𝒚𝟐 ) 𝒂𝒏𝒅 (𝒙𝟑 , 𝒚𝟑 ) are the vertices of a triangle, whose area is 𝒌 square
𝒙𝟏 𝒚𝟏 𝟒𝟐
units, then the value of |𝒙𝟐 𝒚𝟐 𝟒| is [H] [KCET-18]
𝒙𝟑 𝒚𝟑 𝟒
1) 32𝑘 2 2) 16𝑘 2 3) 64𝑘 2 4) 48𝑘 2
𝟑 𝟏
27. The minor of 3 in | | is [E]
𝟓 𝟔
1) 3 2) 1 3) 5 4) 6
𝟐 𝟐 𝟏
28. The cofactor of the element 𝒂𝟏𝟐 in [𝟎 𝟑 𝟑] is [E]
𝟏 −𝟏 𝟒
1) 3 2) −3 3) 4 4) −4
𝒂𝟏𝟏 𝒂𝟏𝟐 𝒂𝟏𝟑
29. If ∆= |𝒂𝟐𝟏 𝒂𝟐𝟐 𝒂𝟐𝟑 | and 𝑨𝒊𝒋 is cofactors of 𝒂𝒊𝒋 , then value of ∆ is given by [E]
𝒂𝟑𝟏 𝒂𝟑𝟐 𝒂𝟑𝟑
1) 𝑎11 𝐴31 + 𝑎12 𝐴32 + 𝑎13 𝐴33 2) 𝑎21 𝐴11 + 𝑎22 𝐴12 + 𝑎23 𝐴13
3) 𝑎11 𝐴11 + 𝑎12 𝐴12 + 𝑎13 𝐴31 4) 𝑎11 𝐴11 + 𝑎21 𝐴21 + 𝑎31 𝐴31
30. If 𝑨 = [𝒂𝒊𝒋 ] is a square matrix of order 3 , then 𝒂𝟏𝟏 𝑨𝟏𝟏 + 𝒂𝟏𝟐 𝑨𝟏𝟐 + 𝒂𝟏𝟑 𝑨𝟏𝟑 is equal to
[E]
1) 0 2) 1 3) |𝐴| 4) −|𝐴|
31. If 𝑨 = [𝒂𝒊𝒋 ] is a square matrix of order 3 , then 𝒂𝟐𝟏 𝑨𝟏𝟏 + 𝒂𝟐𝟐 𝑨𝟏𝟐 + 𝒂𝟐𝟑 𝑨𝟏𝟑 is equal to
[E]
1) 0 2) 1 3) |𝐴| 4) −|𝐴|
32. If A is a 𝟐 × 𝟐 matrix, then |𝒂𝒅𝒋 𝑨| is equal to [E]
1
1) |𝐴|2 2) |𝐴| 3) |𝐴| 4) |𝐴|3

33. If the value of a third order determinant is 16, then the value of the determinant
formed by replacing each of it’s elements by it’s cofactor is [A] [KCET-19]
1) 256 2) 96 3) 16 4) 48

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𝒙 𝒙 −𝟐
34. If cofactor of 𝟐𝒙 in the determinant | 𝟏 𝟐𝒙 𝒙 − 𝟏| is zero, then 𝒙 equal to [A]
𝒙−𝟏 𝒙 𝟎
1) 0 2) 2 3) 1 4) –1
𝟏 𝟐
35. The adjoint of the matrix [ ] is [E]
𝟒 −𝟑
−1 4 −3 −2 −3 4 −3 −4
1) [ ] 2) [ ] 3) [ ] 4) [ ]
2 3 −4 1 2 1 −2 1
36. If A is a 𝟑 × 𝟑 matrix such that |𝟓. 𝒂𝒅𝒋 𝑨| = 𝟓 then |𝑨| is equal to [A] [KCET-22]
1 1
1) ±1 2) ± 5 3) ± 25 4) ±5

37. If A is a square matrix of order 2 and |𝑨| = 𝟒, then |𝑨. 𝒂𝒅𝒋 𝑨| is [A]
1) 4 2) 2 3) 16 4) 64
𝟏 −𝟐 𝟒
38. For a square matrix A, |A| = 6 and 𝒂𝒅𝒋 𝑨 = [ 𝟒 𝟏 𝟏] then 𝒌 = [A]
−𝟏 𝒌 𝟎
2
1) − 3 2) 1 3) –1 4) 2

39. Let A be a matrix of order 𝟑 × 𝟑 and B is it’s adjoint matrix. If |𝑩| = 𝟖𝟏 then |𝑨| =
[A]
1) ±6 2) ±3 3) ±9 4) ±4
40. If A is a matrix of order 2, such that 𝑨(𝒂𝒅𝒋𝑨) = 𝟗𝑰, then |𝒂𝒅𝒋 𝑨| = [A]
1) 9 2) 9I 3) 81 4) 3
𝟏 𝟑
41. If 𝑨 = [ ] be the adjoint of a matrix 𝑨 and |𝑨| = 𝟐, then the value of 𝜶 is
𝟏 𝜶
[A] [KCET-25]
1) 4 2) 5 3) 2 4) 3
𝒂 𝟎 𝟎
42. If 𝑨 = [𝟎 𝒂 𝟎], then 𝒅𝒆𝒕(𝒂𝒅𝒋𝑨) = [A]
𝟎 𝟎 𝒂
1) 𝑎27 2) 𝑎9 3) 𝑎6 4) 𝑎3
𝟏 𝜶 𝟑
43. If 𝑷 = [𝟏 𝟑 𝟑], is the adjoint of a 𝟑 × 𝟑 matrix A and |𝑨| = 𝟒 then 𝜶 is equal to
𝟐 𝟒 𝟒
[A] [KCET-24]
1) 4 2) 5 3) 11 4) 0
𝟐 𝟏 𝟏 𝟎
44. If [ ]𝑨 = [ ] , then 𝑨 = [E] [KCET-20]
𝟑 𝟐 𝟎 𝟏
2 −1 2 1 2 −1 −2 1
1) [ ] 2) [ ] 3) [ ] 4) [ ]
3 2 3 2 −3 2 3 −2
45. If A is an invertible matrix of order 2, then det (𝑨−𝟏) = [E]
1
1) det (A) 2) det 𝐴 3) 1 4) 0

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46. If 𝒅𝒆𝒕 (𝑨) = 𝟒 , then det (𝑨−𝟏 ) = [E]
1
1) 4 2) 4 3) 1 4) 0

47. If A is an invertible matrix of order 3, and |3A| = 3, the value of |adj A| |𝑨−𝟏| is
[D]
1) 3–2 2) 3–10 3) 3–14 4) 3–8
48. If A is a 𝟑 × 𝟑 non singular matrix and if |𝑨| = 𝟑 , then |(𝟐𝑨)−𝟏 | = [A]
1 1
1) 24 2) 24 3) 3 4) 4

49. If 𝑨 is a square matrix of order 𝟑 × 𝟑 and |𝑨| = 𝟑, then the value of |𝟑𝑨−𝟏 | is
[A] [KCET-25]
1
1) 3 2) 3 3) 27 4) 9

50. If the determinant of the adjoint of a matrix of order 3 is 25, then the determinant
of the inverse of the matrix is [A]
1
1) 0.2 2) ±2 3) 5 4) ±0.2
√625

𝟏−𝒏 𝒏
51. If 𝑨𝒏 = [ ] then |𝑨𝟏 | + |𝑨𝟐 | + ⋯ + |𝑨𝟐𝟎𝟐𝟏 | = [D] [KCET-22]
𝒏 𝟏−𝒏
1) −2021 2) (2021)2 3) −(2021)2 4) 4042
52. If A and B are invertible matrices, then which of the following is not
correct? [E]
1) A (adj A) = (adj A) A = A I 2) A (adj A) = (adj A) A = |A|I
3) (AB)−1 = B−1A−1 4) |A| ≠ 0 and |B| ≠ 0 .
𝟐 𝟑
53. The inverse of the matrix [ ] is [E]
𝟒 𝟔
1) [2 3] 2) [ 2 −3] 3) [ 6 −3] 4) Does not exists
4 6 −4 6 −4 2
𝟐 −𝟏
54. If 𝑨 = [ ] , then inverse of the matrix 𝑨𝟑 is [A] [KCET-22]
𝟑 −𝟐
1) 𝐴 2) 𝐼 3) − 𝐼 4) − 𝐴
𝟐 𝟐
55. The inverse of the matrix [ ] is [A]
𝟒 𝟑
1 3 −2 1 −3 2 1 3 2 1 3 2
1) 2 [ ] 2) 2 [ ] 3) 2 [ ] 4) − 2 [ ].
−4 2 4 −2 4 2 4 2
56. If 𝑨𝟐 − 𝟒𝑨 + 𝑰 = 𝟎, then the inverse of 𝑨 is [A]
1) 𝐴 + 𝐼 2) 𝐴 − 4𝐼 3) 𝐴 − 𝐼 4) 4𝐼 − 𝐴
57. If 𝑨 is an invertible matrix of order 𝟐 then |𝑨𝑨−𝟏 | is [E]
1) I 2) 1 3) −1 4) 0

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58. Which of the following is not correct? [E]
1) |𝑎𝑑𝑗 𝐴| = 𝐴𝑛−1 2) |𝑘𝐴| = 𝑘 𝑛−1 |𝐴| 3) (𝐴𝐵)−1 = 𝐵 −1 𝐴−1 4) (𝐴𝐵)′ = 𝐵 ′ 𝐴′
59. For a square matrix 𝑨 in matrix equation 𝑨𝑿 = 𝑩, which of the following is not
correct [E]
1) |𝐴| ≠ 0, there exists a unique solution
2) |𝐴| = 0, and (𝑎𝑑𝑗𝐴)𝐵 ≠ 0, then there exists no solution
3) |𝐴| = 0, and (𝑎𝑑𝑗𝐴)𝐵 = 0, then system may or may not be consistent
4) |𝐴| = 0, and (𝑎𝑑𝑗𝐴)𝐵 = 0, then system may be inconsistent
60. The equations 𝒙 + 𝟐𝒚 + 𝟑𝒛 = 𝟏, 𝟐𝒙 + 𝒚 + 𝟑𝒛 = 𝟐 and 𝟓𝒙 + 𝟓𝒚 + 𝟗𝒛 = 4 have
[A]
1) unique solution 2) many solutions 3) no solution 4) two solution

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

ADDITIONAL PROBLEMS
𝒙 𝟒 𝟑 𝟏
1. If | |=| | then x is equal to [E]
𝒙 𝟖 𝟓 𝟑
1) 1 2) 3 3) 4 4) 5
𝟎
2. Evaluate |𝐜𝐨𝐬 𝟏𝟓𝟎 𝐬𝐢𝐧 𝟏𝟓𝟎 | [A]
𝐬𝐢𝐧 𝟏𝟓 𝐜𝐨𝐬 𝟏𝟓𝟎
1 1 √3
1) 0 2) 2 3) 4) 2
√2

𝟓 + 𝒊 −𝟑𝒊
3. The value of | | is [A]
𝟒𝒊 𝟓−𝒊
1) 12 2) 17 3) 14 4) 24

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𝟑 𝟎 𝟎
4. The value of the determinant |−𝟓 𝟑 𝟎| [E]
𝟏 𝟐 𝟏
1) 9 2) 10 3) 7 4) 0
𝟐 𝟑 −𝟓
5. The value of the determinant ∆= |𝟏 𝟕 −𝟏| is [E]
𝟐 𝟑 −𝟓
1) 1 2) 0 3) 55 4) 2
6. The value of an odd order skew symmetric determinant is [E]
1) perfect square 2) negative 3) ±1 4) 0
7. If |A|=5 then the value of |𝑨′| = [E]
1
1) 1 2) 5 3) 5 4) 25
𝟑 𝟎 𝟐
8. The value of the determinant ∆= |𝟒 𝟎 −𝟏| is [E]
𝟓 𝟎 −𝟓
1) 4 2) 1 3) 0 4) 2
𝟎 𝐬𝐢𝐧 𝜶 − 𝐜𝐨𝐬 𝜶
9. The value of the determinant |− 𝐬𝐢𝐧 𝜶 𝟎 𝐬𝐢𝐧 𝜷 | is [A]
𝐜𝐨𝐬 𝜶 − 𝐬𝐢𝐧 𝜷 𝟎
1) 0 2) 1 3) sin 𝛼 4) cos 𝛼.
𝒌 𝟖
10. Value of 𝒌, for 𝑨 = [ ] which is a singular matrix is [E]
𝟒 𝟐𝒌
1) 4 2) −4 3) ±4 4) 0
11. The area of a triangle with vertices (−𝟑, 𝟎), (𝟑, 𝟎) 𝒂𝒏𝒅 (𝟎, 𝒌) is 𝟗 square units, then
the value of 𝒌 is [E] [KCET-23]
1) 3 2) 9 3) −9 4) 6
𝟏 𝟑 𝟐 −𝟏
12. If 𝑨 = [ ] and 𝑩 = [ ] then |𝑨𝑩𝑩′ | = [E] [KCET-19]
𝟒 𝟐 𝟏 𝟐
1) 100 2) 50 3) 250 4) −250
𝟒𝟒𝟏 𝟒𝟒𝟐 𝟒𝟒𝟑
13. The value of |𝟒𝟒𝟓 𝟒𝟒𝟔 𝟒𝟒𝟕| is [E]
𝟒𝟒𝟏 𝟒𝟒𝟐 𝟒𝟒𝟑
1) 441 × 446 × 443 2) 0 3) −1 4) 1
𝟏 𝟐 𝟐 −𝟏
14. If 𝑨 = [ ] and 𝑩 = [ ] then |𝑨′𝑩𝑩′ | = [E]
𝟒 𝟑 𝟏 𝟑
1) 105 2) 245 3) 25 4) −245
𝟏 𝒕𝒂𝒏 𝜶⁄𝟐
15. If 𝑨 = [ ] and 𝑨𝑩 = 𝑰, then 𝑩 = [H] [KCET-23]
−𝒕𝒂𝒏 𝜶⁄𝟐 𝟏
1) 𝑠𝑖𝑛2 𝛼⁄2 . 𝐴 2) 𝑐𝑜𝑠 2 𝛼⁄2 . 𝐴𝑇 3) 𝑐𝑜𝑠 2 𝛼⁄2 . 𝐴 4) 𝑐𝑜𝑠 2 𝛼⁄2 . 𝐼

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16. If a square matrix A is such that 𝑨𝑨𝑻 = 𝑰, then |𝑨| is equal to [E]
1) 0 2) ±1 3) ±2 4) 4
17. If A and B are square matrices of order 3 such that |𝑨| = −𝟏, |𝑩| = 𝟑 then |𝟑𝑨𝑩|
is equal to [E]
1) −9 2) −27 3) −81 4) 81
18. The area of a triangle with vertices (𝒌, 𝟎), (𝟒, 𝟎) 𝒂𝒏𝒅 (𝟎, 𝟐) is 𝟒 square units, then
the value of 𝒌 is [E]
1) 0 𝑜𝑟 8 2) 0 𝑜𝑟 − 8 3) 0 4) 8
𝟏 𝟑 𝟓
19. The minor of the element 𝒂𝟏𝟑 in |𝟓 𝟐 𝟏| is [E]
𝟎 −𝟏 𝟎
1) 5 2) 3 3) −5 4) −7
𝟑 𝟏
20. The cofactor of 3 in [ ] is [E]
𝟓 𝟔
1) 3 2) 1 3) 5 4) 6
21. If A is a square matrix of order 3 and |𝑨| = 𝟓, then |𝒂𝒅𝒋 𝑨| is [E]
1) 625 2) 5 3) 125 4) 25
22. If 𝑨 = [𝒂𝒊𝒋 ] is a square matrix of order 3 , then 𝒂𝟏𝟏 𝑨𝟑𝟏 + 𝒂𝟏𝟐 𝑨𝟑𝟐 + 𝒂𝟏𝟑 𝑨𝟑𝟑 is equal to [E]
1) 0 2) 1 3) |𝐴| 4) −|𝐴|
23. If A is a matrix of order 3, such that 𝑨(𝒂𝒅𝒋𝑨) = 𝟏𝟎𝐈, then |𝒂𝒅𝒋 𝑨| = [A] [KCET-14]
1)10 2) 10I 3)1 4) 100
24. If A is a square matrix of order 3 and |𝑨| = 𝟓, then |𝑨. 𝒂𝒅𝒋 𝑨| is [A]
1) 625 2) 5 3) 125 4) 25
𝟏𝟎 𝟎
25. For any 2×2 matrix, if A(adjA) = [ ], then |𝐴| is equal to [A]
𝟎 𝟏𝟎
1) 20 2) 100 3) 10 4) 0
𝟐 𝟎 𝟎
26. The inverse of the matrix 𝑨 = [𝟎 𝟑 𝟎] is [A] [KCET-14]
𝟎 𝟎 𝟒
1
0 0
2 0 0 2 2 0 0 1 0 0
1 1 1
1) [0 3 0] 2) 0 3
0 3) 24 [0 3 0] 4) 24 [0 1 0]
0 0 4 1 0 0 4 0 0 1
[0 0 4]

27. If A is a 𝟐 × 𝟐 non singular matrix and if |𝑨| = 𝟓 , then |(𝟓𝑨)−𝟏 | = [A]
1 1
1) 125 2) 125 3) 5 4) 5

28. For a non singular matrix A, |A| is [E]
1) |𝐴| ≥ 0 2) |𝐴| ≤ 0 3) |𝐴| = 0 4) |𝐴| ≠ 0

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29. If A is an invertible matrix of order 2, if det 𝑨 = 𝟓, then det (𝑨−𝟏 ) = [E]
1
1) 25 2) 5 3) 1 4) 125
𝟑 𝟐
30. The inverse of the matrix [ ] is [A]
𝟒 𝟑
3 −2 3 2 −3 −2 3 4
1) [ ] 2) [ ] 3) [ ] 4) [ ]
−4 3 4 3 −4 −3 2 3
𝟐 𝝀 −𝟒
31. If A =[𝟎 𝟐 𝟓 ] , then 𝐀−𝟏 exists if [A]
𝟏 𝟏 𝟑
1) 𝜆 = −2 2) 𝜆 ≠ 2 and 𝜆 ≠ −2 3) 𝜆 ≠ −2 4) 𝜆 ≠ 2
32. If A is an invertible matrix of order 2 and det (𝑨−𝟏) = 6, then det (A) is [E]
1
1) 6 2) 6 3) 36 4) 1

33. If the equation 𝑨𝟐 − 𝟓𝑨 + 𝟕𝑰 = 𝟎, then the inverse of 𝑨 is [A] [KCET-25]
1 1 1 1
1) 7 (5𝐼 − 𝐴) 2) 7 (5𝐼 + 𝐴) 3) 7 (𝐴 − 5𝐼) 4) − 7 (𝐴 + 5𝐼)

34. If 𝑨 is a singular matrix then |𝑨| = 𝟎 and if (𝒂𝒅𝒋𝑨)𝑩 = 𝟎, then system of equations
have [E]
1) unique solution
2) either infinity many solutions or no solution
3) finite solution
4) infinitely many solution
35. The system of equations 𝟒𝒙 + 𝟔𝒚 = 𝟓 and 𝟖𝒙 + 𝟏𝟐𝒚 = 𝟏𝟎 has [E] [KCET-25]
1) no solution 2) infinitely many solutions
3) a unique solution 4) only two solutions
36. Given that A = [aij] is a square matrix of order 3 × 3 and |A| = −7, then the value of
3

 a A , where Aij denotes the cofactor of element a is
i 1
i2 i2 ij [H]

1) 7 2) 0 3) −7 4) 49
𝟏 𝟏 𝟐 |𝐚𝐝𝐣𝐁|
37. If A=[𝟏 𝟑 𝟒], 𝑩 = 𝒂𝒅𝒋 (𝑨) and 𝑪 = 𝟑𝑨 then |𝐂| is [H]
𝟏 −𝟏 𝟑
1) 8 2) 4 3) 2 4) 16
−𝟒 −𝟏
38. If A=[ ], then |𝐀𝟐 |= [A]
𝟑 𝟏
1) 1 2) −1 3) 2 4) −2
𝜶 𝟐
39. If A = [ ] and |𝐀𝟑 |= 27, then 𝜶 = [E]
𝟐 𝜶
1) ±1 2) ±2 3) ±7 4) ±√7

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𝟏 𝜶 𝜶𝟐 𝟑
40. If 𝒇(𝜶) = | 𝜶 𝜶𝟐 𝟏 | , then 𝒇( √𝟑) is equal to [H]
𝜶𝟐 𝟏 𝜶
1) 1 2) 4 3) −4 4) 2

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

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