# Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 - Matrics [Latest edition]

## Chapter 2: Matrics

Exercise 2.1Exercise 2.2Miscellaneous exercise - 2(A)Exercise 2.3Miscellaneous exercise - 2(B)

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Matrics Exercise 2.1 [Pages 39 - 40]

Exercise 2.1 | Q 1 | Page 39

Apply the given elementary transformation of the following matrix.

A = [(1,0),(-1,3)], R1↔ R2

Exercise 2.1 | Q 2 | Page 39

Apply the given elementary transformation of the following matrix.

B = [(1,-1,3),(2,5,4)], R1→ R1 – R2

Exercise 2.1 | Q 3 | Page 39

Apply the given elementary transformation of the following matrix.

A = [(5,4),(1,3)], C1↔ C2; B = [(3,1),(4,5)] R1↔ R2.
What do you observe?

Exercise 2.1 | Q 4 | Page 39

Apply the given elementary transformation of the following matrix.

A = [(1,2,-1),(0,1,3)], 2C2

B = [(1,0,2),(2,4,5)], −3R1

Find the addition of the two new matrices.

Exercise 2.1 | Q 5 | Page 39

Apply the given elementary transformation of the following matrix.

A = [(1,-1,3),(2,1,0),(3,3,1)], 3R3 and then C3 + 2C2

Exercise 2.1 | Q 6 | Page 39

Apply the given elementary transformation of the following matrix.

A = [(1,-1,3),(2,1,0),(3,3,1)], 3R3 and then C3 + 2C2

and A = [(1,-1,3),(2,1,0),(3,3,1)], C3 + 2C2 and then 3R3
What do you conclude.

Exercise 2.1 | Q 7 | Page 39

Apply the given elementary transformation of the following matrix.

Use suitable transformation on [(1,2),(3,4)] to convert it into an upper triangular matrix.

Exercise 2.1 | Q 8 | Page 39

Apply the given elementary transformation of the following matrix.

Convert [(1,-1),(2,3)] into an identity matrix by suitable row transformations.

Exercise 2.1 | Q 9 | Page 40

Apply the given elementary transformation of the following matrix.

Transform [(1,-1,2),(2,1,3),(3,2,4)] into an upper triangular matrix by suitable column transformations.

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Matrics Exercise 2.2 [Pages 51 - 52]

Exercise 2.2 | Q 1.1 | Page 51

Find the co-factor of the element of the following matrix.

[(-1,2),(-3,4)]

Exercise 2.2 | Q 1.2 | Page 51

Find the co-factor of the element of the following matrix.

[(1,-1,2),(-2,3,5),(-2,0,-1)]

Exercise 2.2 | Q 2.1 | Page 51

Find the matrix of the co-factor for the following matrix.

[(1,3),(4,-1)]

Exercise 2.2 | Q 2.2 | Page 51

Find the matrix of the co-factor for the following matrix.

[(1,0,2),(-2,1,3),(0,3,-5)]

Exercise 2.2 | Q 3.1 | Page 51

Find the adjoint of the following matrix.

[(2,-3),(3,5)]

Exercise 2.2 | Q 3.2 | Page 51

Find the adjoint of the following matrix.

[(1,-1,2),(-2,3,5),(-2,0,-1)]

Exercise 2.2 | Q 4 | Page 51

If ∴ verify that A (adj A) = (adj A) A = | A | I

Exercise 2.2 | Q 5.1 | Page 52

Find the inverse of the following matrix by the adjoint method.

[(-1,5),(-3,2)]

Exercise 2.2 | Q 5.2 | Page 52

Find the inverse of the following matrix by the adjoint method.

[(2,-2),(4,3)]

Exercise 2.2 | Q 5.3 | Page 52

Find the inverse of the following matrix by the adjoint method.

[(1,0,0),(3,3,0),(5,2,-1)]

Exercise 2.2 | Q 5.4 | Page 52

Find the inverses of the following matrices by the adjoint method:

[(1,2,3),(0,2,4),(0,0,5)]

Exercise 2.2 | Q 6.1 | Page 52

Find the inverse of the following matrix.

[(1,2),(2,-1)]

Exercise 2.2 | Q 6.2 | Page 52

Find the inverse of the following matrix.

[(2,-3),(-1,2)]

Exercise 2.2 | Q 6.3 | Page 52

Find the inverse of the following matrix.

[(0,1,2),(1,2,3),(3,1,1)]

Exercise 2.2 | Q 6.4 | Page 52

Find the inverse of the following matrix.

[(2,0,-1),(5,1,0),(0,1,3)]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Matrics Miscellaneous exercise - 2(A) [Pages 52 - 54]

Miscellaneous exercise - 2(A) | Q 1 | Page 52

If A = ((1,0,0),(2,1,0),(3,3,1)), then reduce it to I3 by using column transformations.

Miscellaneous exercise - 2(A) | Q 2 | Page 52

If A = ((2,1,3),(1,0,1),(1,1,1)), then reduce it to I3 by using row transformations.

Miscellaneous exercise - 2(A) | Q 3.1 | Page 52

Check whether the following matrix is invertible or not:

((1,0),(0,1))

Miscellaneous exercise - 2(A) | Q 3.2 | Page 52

Check whether the following matrix is invertible or not:

((1,1),(1,1))

Miscellaneous exercise - 2(A) | Q 3.3 | Page 52

Check whether the following matrix is invertible or not:

((1,2),(3,3))

Miscellaneous exercise - 2(A) | Q 3.4 | Page 52

Check whether the following matrix is invertible or not:

((2,3),(10,15))

Miscellaneous exercise - 2(A) | Q 3.5 | Page 52

Check whether the following matrix is invertible or not:

(("cos" theta , "sin" theta),(- "sin" theta,"cos" theta))

Miscellaneous exercise - 2(A) | Q 3.6 | Page 52

Check whether the following matrix is invertible or not:

(("sec" theta , "tan" theta),("tan" theta,"sec" theta))

Miscellaneous exercise - 2(A) | Q 3.7 | Page 52

Check whether the following matrix is invertible or not:

((3,4,3),(1,1,0),(1,4,5))

Miscellaneous exercise - 2(A) | Q 3.8 | Page 52

Check whether the following matrix is invertible or not:

((1,2,3),(2,-1,3),(1,2,3))

Miscellaneous exercise - 2(A) | Q 3.9 | Page 52

Check whether the following matrix is invertible or not:

((1,2,3),(3,4,5),(4,6,8))

Miscellaneous exercise - 2(A) | Q 4 | Page 52

Find AB, if A = ((1,2,3),(1,-2,-3)) and B = ((1,-1),(1,2),(1,-2)). Examine whether AB has inverse or not.

Miscellaneous exercise - 2(A) | Q 5 | Page 52

If A = [("x",0,0),(0,"y",0),(0,0,"z")] is a non-singular matrix, then find A-1 by using elementary row transformations. Hence, find the inverse of [(2,0,0),(0,1,0),(0,0,-1)].

Miscellaneous exercise - 2(A) | Q 6 | Page 53

If A = [(1,2),(3,4)] and X is a 2 × 2 matrix such that AX = I, find X.

Miscellaneous exercise - 2(A) | Q 7.01 | Page 53

Find the inverse of the following matrix (if they exist):

((1,-1),(2,3))

Miscellaneous exercise - 2(A) | Q 7.02 | Page 53

Find the inverse of the following matrix (if they exist):

((2,1),(1,-1))

Miscellaneous exercise - 2(A) | Q 7.03 | Page 53

Find the inverse of the following matrix (if they exist):

((1,3),(2,7))

Miscellaneous exercise - 2(A) | Q 7.04 | Page 53

Find the inverse of the following matrix (if they exist):

[(2,-3),(5,7)]

Miscellaneous exercise - 2(A) | Q 7.05 | Page 53

Find the inverse of the following matrix (if they exist):

[(2,1),(7,4)]

Miscellaneous exercise - 2(A) | Q 7.06 | Page 53

Find the inverse of the following matrix (if they exist):

[(3,-10),(2,-7)]

Miscellaneous exercise - 2(A) | Q 7.07 | Page 53

Find the inverse of the following matrix (if they exist):

[(2,-3,3),(2,2,3),(3,-2,2)]

Miscellaneous exercise - 2(A) | Q 7.08 | Page 53

Find the inverse of the following matrix (if they exist).

[(1,3,-2),(-3,0,-5),(2,5,0)]

Miscellaneous exercise - 2(A) | Q 7.09 | Page 53

Find the inverse of the following matrix (if they exist):

[(2,0,-1),(5,1,0),(0,1,3)]

Miscellaneous exercise - 2(A) | Q 7.1 | Page 54

Find the inverse of the following matrix by elementary row transformations if it exists. A=[[1,2,-2],[0,-2,1],[-1,3,0]]

Miscellaneous exercise - 2(A) | Q 8.1 | Page 53

Find the inverse of A = [("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)] by elementary row transformations.

Miscellaneous exercise - 2(A) | Q 8.2 | Page 53

Find the inverse of A = [("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)] by elementary column transformations.

Miscellaneous exercise - 2(A) | Q 9 | Page 53

If A = [(2,3),(1,2)], B = [(1,0),(3,1)], find AB and (AB)-1 . Verify that (AB)-1 = B-1.A-1.

Miscellaneous exercise - 2(A) | Q 10 | Page 53

If A = [(4,5),(2,1)], show that "A"^-1 = 1/6("A" - 5"I").

Miscellaneous exercise - 2(A) | Q 11 | Page 53

Find the matrix X such that AX = B, where A = [(1,2),(-1,3)] and B = [(0,1),(2,4)]

Miscellaneous exercise - 2(A) | Q 12 | Page 53

Find X, if AX = B, where A = [(1,2,3),(-1,1,2),(1,2,4)] and B = [(1),(2),(3)]

Miscellaneous exercise - 2(A) | Q 13 | Page 54

If A = [(1,1),(1,2)], "B" = [(4,1),(3,1)] and C = [(24,7),(31,9)], then find the matrix X such that AXB = C.

Miscellaneous exercise - 2(A) | Q 14 | Page 54

Find the inverse of [(1,2,3),(1,1,5),(2,4,7)] by the adjoint method.

Miscellaneous exercise - 2(A) | Q 15 | Page 54

Find the inverse of matrix A by using adjoint method; where A = [(1, 0, 1), (0, 2, 3), (1, 2, 1)]

Miscellaneous exercise - 2(A) | Q 16 | Page 54

Find A-1 by the adjoint method and by elementary transformations, if A = [(1,2,3),(-1,1,2),(1,2,4)]

Miscellaneous exercise - 2(A) | Q 17 | Page 54

Find the inverse of A = [(1,0,1),(0,2,3),(1,2,1)] by using elementary column transformations.

Miscellaneous exercise - 2(A) | Q 18 | Page 54

Find the inverse of [(1,2,3),(1,1,5),(2,4,7)] by using elementary row transformations.

Miscellaneous exercise - 2(A) | Q 19.1 | Page 54

Show with the usual notation that for any matrix A = ["a"_"ij"]_(3xx3)  "is"   "a"_11"A"_21 + "a"_12"A"_22 + "a"_13"A"_23 = 0

Miscellaneous exercise - 2(A) | Q 19.2 | Page 54

Show with the usual notation that for any matrix A = ["a"_"ij"]_(3xx3)  "is"   "a"_11"A"_11 + "a"_12"A"_12 + "a"_13"A"_13 = |"A"|

Miscellaneous exercise - 2(A) | Q 20 | Page 54

If A = [(1,0,1),(0,2,3),(1,2,1)] and B = [(1,2,3),(1,1,5),(2,4,7)], then find a matrix X such that XA = B.

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Matrics Exercise 2.3 [Pages 59 - 60]

Exercise 2.3 | Q 1.1 | Page 59

Solve the following equations by inversion method.

x + 2y = 2, 2x + 3y = 3

Exercise 2.3 | Q 1.2 | Page 59

Solve the following equations by inversion method:

x + y = 4, 2x - y = 5

Exercise 2.3 | Q 1.3 | Page 59

Solve the following equations by inversion method.

2x + 6y = 8, x + 3y = 5

Exercise 2.3 | Q 2.1 | Page 60

Solve the following equations by the reduction method.

2x + y = 5, 3x + 5y = – 3

Exercise 2.3 | Q 2.2 | Page 60

Solve the following equations by the reduction method.

x + 3y = 2, 3x + 5y = 4

Exercise 2.3 | Q 2.3 | Page 60

Solve the following equations by the reduction method.

3x – y = 1, 4x + y = 6

Exercise 2.3 | Q 2.4 | Page 60

Solve the following equations by the reduction method.

5x + 2y = 4, 7x + 3y = 5

Exercise 2.3 | Q 3 | Page 60

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Exercise 2.3 | Q 4 | Page 60

If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices.

Exercise 2.3 | Q 5 | Page 60

The total cost of 3 T.V. sets and 2 V.C.R.’s is ₹ 35,000. The shopkeeper wants a profit of ₹ 1000 per T.V. set and ₹ 500 per V.C.R. He sells 2 T.V. sets and 1 V.C.R. and gets the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. set and a V.C.R.

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Matrics Miscellaneous exercise - 2(B) [Pages 61 - 63]

Miscellaneous exercise - 2(B) | Q 1.01 | Page 61

Choose the correct answer from the given alternatives in the following question:

If A = [(1,2),(3,4)] , adj A = [(4,"a"),(-3,"b")], then the values of a and b are

• a = - 2, b = 1

• a = 2, b = 4

• a = 2, b = - 1

• a = 1, b = - 2

Miscellaneous exercise - 2(B) | Q 1.02 | Page 61

Choose the correct answer from the given alternatives in the following question:

The inverse of [(0,1),(1,0)] is

• [(1,1),(1,1)]

• [(0,1),(1,0)]

• [(1,0),(0,1)]

• none of these

Miscellaneous exercise - 2(B) | Q 1.03 | Page 61

Choose the correct answer from the given alternatives in the following question:

If A = [(1,2),(2,1)] and A(adj A) = k I, then the value of k is

• 1

• -1

• 0

• -3

Miscellaneous exercise - 2(B) | Q 1.04 | Page 61

Choose the correct answer from the given alternatives in the following question:

If A = [(2,-4),(3,1)], then the adjoint of matrix A is

• [(-1,3),(-4,1)]

• [(1,4),(-3,2)]

• [(1,3),(4,-2)]

• [(-1,-3),(-4,2)]

Miscellaneous exercise - 2(B) | Q 1.05 | Page 62

Choose the correct answer from the given alternatives in the following question:

If A = [(1,2),(3,4)], and A (adj A) = kI, then the value of k is

• 2

• - 2

• 10

• - 10

Miscellaneous exercise - 2(B) | Q 1.06 | Page 62

Choose the correct answer from the given alternatives in the following question:

If A = [(lambda,1),(-1, -lambda)], and A-1 does not exist if λ = _______

• 0

• ± 1

• 2

• 2

Miscellaneous exercise - 2(B) | Q 1.07 | Page 62

Choose the correct answer from the given alternatives in the following question:

If A = [("cos"alpha, "sin"alpha),("sin"alpha,"cos"alpha)], then A-1 = _____

• [(1/("cos"alpha),-1/("sin" alpha)),(1/("sin"alpha),1/("cos"alpha))]

• [("cos"alpha,-"sin"alpha),(-"sin"alpha, "cos"alpha)]

• [(-"cos"alpha,"sin"alpha),(-"sin"alpha, "cos"alpha)]

• [(-"cos"alpha,"sin"alpha),("sin"alpha, -"cos"alpha)]

Miscellaneous exercise - 2(B) | Q 1.08 | Page 62

Choose the correct answer from the given alternatives in the following question:

If A = [("cos"alpha, - "sin"alpha,0),("sin"alpha,"cos"alpha,0),(0,0,1)] where α ∈ R, then [F(α)]-1 is

• F(- α)

• F(α-1)

• F(2α)

• none of these

Miscellaneous exercise - 2(B) | Q 1.09 | Page 62

Choose the correct answer from the given alternatives in the following question:

The inverse of A = [(0,1,0),(1,0,0),(0,0,1)] is

• 1

• A

• A'

• - I

Miscellaneous exercise - 2(B) | Q 1.1 | Page 63

Choose the correct answer from the given alternatives in the following question:

The inverse of a symmetric matrix is

• symmetric

• non-symmetric

• null matrix

• diagonal matrix

Miscellaneous exercise - 2(B) | Q 1.11 | Page 63

Choose the correct answer from the given alternatives in the following question:

For a 2 × 2 matrix A, if A(adj A) = [(10,0),(0,10)], then determinant A equals

• 20

• 10

• 30

• 40

Miscellaneous exercise - 2(B) | Q 1.12 | Page 63

Choose the correct answer from the given alternatives in the following question:

If A' = - 1/2[(1,-4),(-1,2)], then A = ______.

• [(2,4),(-1,1)]

• [(2,4),(1,-1)]

• [(2,-4),(1,1)]

• [(2,4),(1,1)]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 2 Matrics Miscellaneous exercise - 2(B) [Page 63]

Miscellaneous exercise - 2(B) | Q 1.1 | Page 63

Solve the following equation by the method of inversion:

2x - y = - 2, 3x + 4y = 3

Miscellaneous exercise - 2(B) | Q 1.2 | Page 63

Solve the following equations by the method of inversion:

x + y+ z = 1, 2x + 3y + 2x = 2, ax + ay + 2az = 4, a ≠ 0.

Miscellaneous exercise - 2(B) | Q 1.3 | Page 63

Solve the following equations by the method of inversion:

5x - y + 4z = 5, 2x + 3y + 5z = 2, 5x - 2y + 6z = - 1

Miscellaneous exercise - 2(B) | Q 1.4 | Page 63

Solve the following equations by the method of inversion:

2x - y = - 2, 3x + 4y = 3

Miscellaneous exercise - 2(B) | Q 1.5 | Page 63

Solve the following equations by the method of inversion:

x + y + z = - 1, y + z = 2, x + y - z = 3

Miscellaneous exercise - 2(B) | Q 2.1 | Page 63

Express the following equations in matrix form and solve them by the method of reduction:

x - y + z = 1, 2x - y = 1, 3x + 3y - 4z = 2

Miscellaneous exercise - 2(B) | Q 2.2 | Page 63

Express the following equations in matrix form and solve them by the method of reduction:

x + y = 1, y + z = 3, z + x = 3.

Miscellaneous exercise - 2(B) | Q 2.3 | Page 63

Express the following equations in matrix form and solve them by the method of reduction:

2x - y + z = 1, x + 2y + 3x = 8, 3x + y - 4z = 1.

Miscellaneous exercise - 2(B) | Q 2.5 | Page 63

Express the following equations in matrix form and solve them by the method of reduction:

x + 2y + z = 8, 2x + 3y - z = 11, 3x - y - 2z = 5.

Miscellaneous exercise - 2(B) | Q 2.6 | Page 63

Express the following equations in matrix form and solve them by the method of reduction:

x + 3y + 2z = 6, 3x - 2y + 5z = 5, 2x - 3y + 6z = 7

Miscellaneous exercise - 2(B) | Q 3 | Page 63

The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number we get 11. By adding first and third numbers we get a number, which is double than the second number. Use this information and find a system of linear equations. Find these three numbers using matrices.

Miscellaneous exercise - 2(B) | Q 4 | Page 63

The cost of 4 pencils, 3 pens, and 2 books is ₹ 150. The cost of 1 pencil, 2 pens, and 3 books is ₹ 125. The cost of 6 pencils, 2 pens, and 3 books is ₹ 175. Find the cost of each item by using matrices.

Miscellaneous exercise - 2(B) | Q 5 | Page 63

The sum of three numbers is 6. Thrice the third number when added to the first number, gives 7. On adding three times the first number to the sum of second and third numbers, we get 12. Find the three number by using matrices.

Miscellaneous exercise - 2(B) | Q 6 | Page 63

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Miscellaneous exercise - 2(B) | Q 7 | Page 63

An amount of ₹ 5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from these investments is ₹ 350. If the total annual income from the first two investments is ₹ 70 more than the income from the third, find the amount of each investment using matrix method.

## Chapter 2: Matrics

Exercise 2.1Exercise 2.2Miscellaneous exercise - 2(A)Exercise 2.3Miscellaneous exercise - 2(B)

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 - Matrics

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