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## Chapter 2: Matrics

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

Apply the given elementary transformation of the following matrix.

A = `[(1,0),(-1,3)]`, R_{1}↔ R_{2}

Apply the given elementary transformation of the following matrix.

B = `[(1,-1,3),(2,5,4)]`, R_{1}→ R_{1} – R_{2}

Apply the given elementary transformation of the following matrix.

A = `[(5,4),(1,3)]`, C_{1}↔ C_{2}; B = `[(3,1),(4,5)]` R_{1}↔ R_{2}.

What do you observe?

Apply the given elementary transformation of the following matrix.

A = `[(1,2,-1),(0,1,3)]`, 2C_{2}

B = `[(1,0,2),(2,4,5)]`, −3R_{1}

Find the addition of the two new matrices.

Apply the given elementary transformation of the following matrix.

A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R_{3} and then C_{3} + 2C_{2}

Apply the given elementary transformation of the following matrix.

A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R_{3} and then C_{3} + 2C_{2}

and A = `[(1,-1,3),(2,1,0),(3,3,1)]`, C_{3} + 2C_{2} and then 3R_{3}What do you conclude._{}

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.

Apply the given elementary transformation of the following matrix.

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

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 MatricsExercise 2.2[Pages 51 - 52]

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

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

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

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

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

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

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

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

Find the adjoint of the following matrix.

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

Find the adjoint of the following matrix.

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

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

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

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

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

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

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

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

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

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

Find the inverse of the following matrix.

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

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 MatricsMiscellaneous exercise 2 (A)[Pages 52 - 54]

If A = `((1,0,0),(2,1,0),(3,3,1))`, then reduce it to I_{3} by using column transformations.

If A = `((2,1,3),(1,0,1),(1,1,1))`, then reduce it to I_{3} by using row transformations.

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

**Check whether the following matrix is invertible or not:**

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

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.

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)]`

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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`

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"|`

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 MatricsExercise 2.3[Pages 59 - 60]

Solve the following equations by inversion method.

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

**Solve the following equations by inversion method:**

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

Solve the following equations by inversion method.

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

Solve the following equations by the reduction method.

2x + y = 5, 3x + 5y = – 3

Solve the following equations by the reduction method.

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

Solve the following equations by the reduction method.

3x – y = 1, 4x + y = 6

Solve the following equations by the reduction method.

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

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.

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.

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 MatricsMiscellaneous exercise 2 (B)[Pages 61 - 63]

**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

**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

**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

**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)]`

**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

**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

**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)]`

**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

**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

**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

**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

**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 MatricsMiscellaneous exercise 2 (B)[Page 63]

**Solve the following equation by the method of inversion:**

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

**Solve the following equations by the method of inversion:**

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

**Solve the following equations by the method of inversion:**

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

**Solve the following equations by the method of inversion:**

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

**Solve the following equations by the method of inversion:**

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

**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

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

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

**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.

**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.

**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

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.

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.

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.

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.

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

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

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 2 Matrics are Elementry Transformations, Inverse of a Matrix, Application of Matrices, Elementary Operation (Transformation) of a Matrix, Inverse of Matrix, Applications of Determinants and Matrices.

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