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ASSIGNMENT-2
Mathematics NCERT 12th CBSE pattern-----------------------------------PAPER CODE: MA3-AST-2
Chapter 3 – Matrices
General Instructions
Each question in Section A is of 1 mark and Each Question of Section B is of four marks.
Section A
Q1. If matrix 𝐴 = [1 2 3], write matrix 𝐴𝐴′ where 𝐴′ is the transpose of matrix 𝐴.
Q2. Fill in the blanks
i. (𝐴 + 𝐵 )′ = ___________________________________
ii. (𝐴 − 𝐵 )′ = ___________________________________
iii. (𝐴𝐵 )′ = ___________________________________
iv. (𝑘𝐴 )′ = ___________________________________
v. (𝐴′ )′ = ___________________________________
Q3. Define a symmetric matrix and a skew-symmetric matrix. Give an example of each.
Q4. If A is a square matrix of order n, then 𝐴 + 𝐴′ is a symmetric matrix or a skew-symmetric
matrix.
Q5. If A is a square matrix of order n , then 𝐴 − 𝐴′ is a symmetric matrix or a skew-symmetric
matrix.
Q6. Give an example of a matrix of order 3 which is symmetric as well as skew-symmetric matrix.
1 4
], then show that 𝐴 − 𝐴′ is a skew-symmetric matrix, where 𝐴′ is the transpose
3 7
of matrix 𝐴.
Q7. If 𝐴 = [
Q8. If 𝐴 and 𝐵 are skew-symmetric matrices of the same order, prove that 𝐴𝐵 is symmetric iff 𝐴
and 𝐵 commute.
Q9. If 𝐴 and 𝐵 are symmetric matrices of the same order, prove that
i. 𝐴𝐵 − 𝐵𝐴 is a skew-symmetric matrix
ii. 𝐴𝐵 + 𝐵𝐴 is a symmetric matrix
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−2
Q10. If the matrix 𝐴 = [ 1
𝑥+𝑦
𝑥−𝑦
0
𝑧
5
4] is symmetric matrix, find the values of 𝑥 , 𝑦 𝑎𝑛𝑑 𝑧
7
0 𝑎
Q11. If the matrix 𝐴 = [2 𝑏
𝑐 1
5
−1] is skew-symmetric matrix, find the values of 𝑎 , 𝑏 𝑎𝑛𝑑 𝑐.
0
Q12. Find the values of 𝑦 − 𝑥 for the following equation
𝑥
2[
7
5
3
]+[
𝑦−3
1
7
6
−4
]= [
]
2
15 14
15 𝑥 + 𝑦
15
Q13. If [
]= [
𝑥−𝑦
2
𝑦
𝑐𝑜𝑠 𝛼
Q14. If 𝐴 = [
sin 𝛼
8
], find the value of 𝑥.
3
− sin 𝛼
], then for what value of 𝛼, 𝐴 is an identity matrix?.
cos 𝛼
Q15. Write the value of 𝑥 − 𝑦 + 𝑧 from the following equation:
𝑥+𝑦+𝑧
9
𝑥
+
𝑧
[
] = [ 5]
𝑦+𝑧
7
Q16. If 𝐴 is a matrix of order 3x4 and 𝐵 is a matrix of order 4x3, find the order of matrix(𝐴𝐵)′.
𝑎
Q17. Find the value of a and b such that matrix 𝐴 = [−2
3
−2
3 −2
Q18. If 𝐴 = [
], find 𝑘 such that 𝐴2 = 𝑘𝐴 − 2𝐼2
4 −2
2 3
Q19. If 𝐴 = [
], write 𝐴−1 in terms of 𝐴.
5 −2
3 −4
Q20. If 𝐴 = [
], find a matrix 𝐵 such that 𝐴𝐵 = 𝐼
−1 2
1
−2 −2
−1 𝑏 ] which satisfies 𝐴𝐴′ = 𝐼
𝑏 −1
Section B
Q21. Using elementary transformations, find the inverse of the matrix [
2 −3
Q22. a. Express [6 2
1 5
result.
4 3
]
2 4
4
0] as the sum of symmetric and skew-symmetric matrix and verify the
1
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2 2
b. Express [4 1
0 6
result.
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5
3] as the sum of symmetric and skew-symmetric matrix and verify the
7
4 2 −1
c. Express [3 5
7 ] as the sum of symmetric and skew-symmetric matrix and verify the
1 −2 1
result.
1 2
d. Express [ 4 5
−1 2
result.
3
6] as the sum of symmetric and skew-symmetric matrix and verify the
0
Q23. For any square matrix A, prove that AA’ and A’A are symmetric. Also verify the result for the
matrix
0 2 1
[ 3 0 4]
2 1 0
Q24. Using elementary transformations, find the inverse of the following matrices (if exist)
0 1 2
i. [1 2 3]
3 1 1
2 −3 5
ii. [3 2 −4]
1 1 −2
2 −1 4
iii. [4 0 2]
3 −2 7
1
2 −2
iv. [−1 3
0]
0 −2 1
v. [
2 5
]
1 3
−5
vi. [
2
3
]
2
Q25. If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is
a zero matrix. Justify your answer.
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Q26. A square matrix A satisfies 𝐴2 = 𝐼 − 𝐴, where 𝐼 is the identity matrix. If 𝐴𝑛 = 5𝐴 − 3𝐼. Find
the value of n.
Q27. Using Principle of Mathematical Induction if A be a square matrix such that 𝐴2 = 𝐴, then
show that (𝐼 + 𝐴)𝑛 = 𝐼 + (2𝑛 − 1)𝐴, ∀ 𝑛 ∈ 𝑁.
Q28. Prove that inverse of every square matrix, if exits, is unique.
Q29. Show that all diagonal elements of a skew symmetric matrix is zero.
Q30. Prove that every square matrix can be uniquely expressed as the sum of a symmetric and a
skew-symmetric matrix.
Q31. Show that 𝐵′ 𝐴𝐵 is symmetric or skew-symmetric matrix according as A is symmetric or
skew-symmetric.
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