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Topic: Matrices
Question bank with solutions
One mark question ( V S A)
- Define matrix
- Define a diagonal matrix
- Define scalar matrix
- Define symmetric matrix
- Define skew-symmetric matrix
6.In a matrix (^) [
5 2 12 โ3^1 โ5 17
]
find 1) order of the matrix
- Write the elements of ๐ 13 , ๐ 21 , ๐ 33 , ๐ 24 , ๐ 23
- If a matrix 8 elements what is the possible order it can have?
- If a matrix 18 elements what is the possible order it can have?
- construct 2 ร 2 matrix [๐๐๐] whose elements are given by
(๐+๐) 2 2
- construct the 2 ร 3 matrix whose elements are given by ๐๐๐= |๐ โ ๐|
- Construct the 3ร 2 matrix whose elements are given by ๐๐๐=
๐ ๐
- Find x, y, z if [^4 ๐ฅ 5
]= [
]
- Find x, y, z if [
5 + ๐ง ๐ฅ๐ฆ]=^ [
]
- Find the matrix x such that 2A + B + X =0 where A = [โ1^2 3 4
] and B = [^3 โ 1 5
]
- If A = [
] B = [
] Find 2A โ B
- Find X if Y =[
] and 2X+Y = [
]
- Find X If X+Y = [^7 2 5
] and X-Y = [^3 0 3
]
- Simplify cos ๐ [ cos ๐ sin ๐ โsin ๐ cos ๐
] + sin ๐ [ sin ๐ โcos ๐ cos ๐ sin ๐
]
- Find X If 2[
] + [
] = [
]
- If A = [^1 3 4
] Find A + ๐ด^1
21. A = [
] and B = [^0 2 6 โ3 1
] Find 3A + 2B
- if A = [ sin ๐^ cos ๐ โcos ๐ sin ๐
] Verify A A^1 = I
- if B = [ cos ๐ sin ๐ โsin ๐ cos ๐
] verify B B^1 = I
- If A = [
] B = [1 5 7]^ Find AB
- Compute 1) [^1 โ 2 3
] [^1 2
]
2) [
] [
]
- Find X and Y [
] = [^6
]
- What is the number of possible square matrix order 3 with each entries 0 or 1
- Find X and Y if [
] is a scalar matrix
- Find X [
] is a symmetric matrix
- By using elementary operation Find the inverse of the matrix [^1 2 7
]
- By using elementary operation Find the inverse of the matrix [^1 โ 2 1
]
- Find P-1^ if it exists and P = [
]
- If A = [^3 โ1 2
] Show that A^2 -5A +7I = 0
- If A = [
] and B = [
] Show that (AB)^1 = B^1 A^1
III. Five mark questions ( LA)
1.If A = [
] B = [
] and C =[
]
Find A B , BC and show that (AB )C = A(BC)
- If A = [
] B = [
] C = [
] calculate AC, BC and (A+B) C
Deduce that (A+B) C = AC + BC
- If A = [
] Show that A^3 โ 23A - 40 I = 0
- If A = [
] B = [
] and C = [
]
verify A+ (B-C) = (A+B ) โ C
- If A =
[
2 3 1
5 3 1 3
2 3
4 3 7 3 2
2 3 ]
and B =
[
2 5
3 5 1 1 5
2 5
4 5 7 5
6 5
2 5 ]
find 3A โ 5B
- If A = [
] find A^2 โ 5 A + 6 I?
- If A = [
] prove that A^3 โ 6A^2 + 7A + 2I = 0
- Express the matrix B = [
] Find the sum of symmetric and skew-
symmetric matrix
- Express the matrix B = [
] Find the sum of symmetric and skew-
symmetric matrix
- If A = [
] B = [
] C = [
] calculate AB , BC, A(B+C)
Verify that AB + AC = A(B+C)
- If F(x) = [
cos ๐ฅ โ sin ๐ฅ 0 sin ๐ฅ cos ๐ฅ 0 0 0 1
] show that F(x) F(y) = F(x+y)
- If A =[
] and B = [1 3 โ6] verify (AB)^1 = B^1 A^1
- If A = [ cos ๐^ sin ๐ โsin ๐ cos ๐
] Prove that An^ = [ cos ๐๐^ sin ๐๐ โsin ๐๐ cos ๐๐
]
21. 3 A +2 B = [^3 โ2^19
]
- A A^1 = [ sin ๐^ cos ๐ โcos ๐ sin ๐
] [sin ๐^ โcos ๐ cos ๐ sin ๐
] = [^1
] = I after multiplying
- BB 1 = [ cos ๐ sin ๐ โsin ๐ cos ๐
] [
cos ๐ โsin ๐ sin ๐ cos ๐
] = [
]= I after multiplying
24. (AB)^1 = [
]
25. 1) [โ3^ โ4^1
] 2) [^14 โ
] after multiplying
- 3 |๐ด| = K |๐ด| implies K = 3
- Y = 0, X = 3 by solving
- The square matrix of order 3X3 = 9 and 2 entries
Then possible entries is 2^9 = 512
29. [5 โ ๐ฅ^ 2๐ฆ โ 8
]= [^3
] then X = 2 Y = 4
- [ 4 ๐ฅ + 2 2๐ฅ โ 3 ๐ฅ + 1
] = [ 4 2๐ฅ โ 3
] implies X = 5
Solutions : Two mark and Three marks questions (SA)
- books pens
Radha : 15 6 this can be expressed as [
] or [^15 10 6 2 5
]
Fauzia : 10 2 Simran: 13 5
[
5 2 1 2 2 0 3 2 ]
- X+ Y + Z = 9 X +Z = 5 Y + Z = 7
7 + Z = 9 X + 2 = 5 Y + 2 = 7
Z= 2 X = 3 Y = 5
- By solving equality a =1 , b= 2, c =3 and d = 4
5. X =
[
10 3 4 14 3 โ 31 3
โ 3 ]
- compare two matrices X = 2, Y = 9
- by solving we get X = 3 , Y = -
- by solving and compare we get X = 2 , Y = 4, Z = 1, w = 3
9. ๐ด๐ ๐ด๐ = [
cos ๐ฅ sin ๐ฅ โsin ๐ฅ cos ๐ฅ
] [
cos ๐ฆ sin ๐ฆ โsin ๐ฆ cos ๐ฆ
] = [
cos(๐ฅ + ๐ฆ) sin(x + ๐ฆ) โsin(๐ฅ + ๐ฆ) cos(๐ฅ + ๐ฆ)
] = ๐ด๐+๐
10. A^2 = KA โ 2I
[
] = [
]
Then 4K = 4
K = 1
11. ( A+B)^1 = [
โ3 โ 1^4
] and A^1 + B 1 = [
โ3 โ 1^4
]
Hence ( A+B)^1 = A^1 + B 1
- B = A + A^1 , B^1 = (A + A^1 )^1 = A^1 +A = B โด B = A +A^1 is symmetric
C = A โ A 1 , C 1 = (A -A^1 )^1 = A^1 -A = -(A- A^1 ) = - C โด C = A โ A^1 is skew- symmetric
13. Z = A + A^1 = [
] + [
] = [
] = Z 1 โด Z = Z^1 = A + A^1 is symmetric
14. Z^1 = (A - A^1 )^1 = ([
] โ [
])
1 = ([
] )
1 = -Z โด Z^1 = - Z, A โ A^1
skew- symmetric
- (AB) (AB)-1^ = I
A-1(AB) (AB)-1^ = A-1I I A = A B(AB)-1^ = A-1^ IA-1^ = A- B-1B(AB)-1^ = B-1A-1^ AA-1^ = I (AB)-1^ = B-1A-1^ BB-1^ I
Solutions : Five mark questions (LA)
1.AB =[
] (AB) C = [
]
BC = [
] A(BC) = [
]
Hence (AB) C = A(BC)
2. (A +B) C = [
] AC = [
] BC = [
] AC + BC = [
]
Hence (A +B) C = AC + BC
3. A = [
] A^2 = [
] A^3 = [
]
LHS = A3 โ 23A โ 40 I = 0 By simplification
4. A + (B โ C) = [
] and (A+B) โ C = [
]
Hence A + (B โ C) = (A+B) โ C
5. 3A -5B = [
] - [
] = [
] = 0
6. A^2 โ 5A + 6I = [
] by simplification
- If A = [
] by calculating A^2 , A^3 take LHS = RHS
8. B = [
] by theorem number 2
B =
1 2 (B +B^
2 (B -B^
(^1) ) hence they are equal
9. B = [
] by theorem number 2
B =
1 2 (B +B^
2 (B -B^
(^1) ) hence they are equal
- If AB = [
] AC = [
] A(B+C) = [
] = AB + AC
- F(x).F(y) = [
cos ๐ฅ โ sin ๐ฅ 0 sin ๐ฅ cos ๐ฅ 0 0 0 1
] [
cos ๐ฆ โ sin ๐ฆ 0 sin ๐ฆ cos ๐ฆ 0 0 0 1
]
= [
cos(๐ฅ + ๐ฆ) โ sin(๐ฅ + ๐ฆ) 0 sin(๐ฅ + ๐ฆ) cos(๐ฅ + ๐ฆ) 0 0 0 1
] = F(x+y)
13. LHS = (AB)^1 = [
] = B^1 A^1 = RHS
- By mathematical induction we get the solution
