# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Matrices [Latest edition]

## Chapter 1: Matrices

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Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.1

#### MCQ [1 Mark]

Q.1 | Q 1

Choose the correct alternative:

If A = [(6, 0),("p", "q")] is a scalar matrix, then the values of p and q are ______ respectively.

• 6 and 0

• 0 and 6

• 5 and 6

• 0 and 1

Q.1 | Q 2

Choose the correct alternative:

If B = [(6, 3),(-2, "k")] is singular matrix, then the value of k is ______

• – 1

• 2

• 0

• 1

Q.1 | Q 3

Choose the correct alternative:

If A = [(1, 3/5, x),(y, -5, -7),(-4, -7, 0)] is a symmetric matrix, then the values of x and y are ______ respectively.

• 3/5 and 4

• 5/3 and – 4

• 3/5 and – 4

• – 4 and 3/5

Q.1 | Q 4

Choose the correct alternative:

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

• [2]

• 1

• [1]

• 2

Q.1 | Q 5

Choose the correct alternative:

If A = [(2, 0),(0, 2)], then A2 – 3I = ______

• O

• I

• A

• 3A

Q.1 | Q 6

Choose the correct alternative:

For any square matrix B, matrix B + BT is ______

• Null matrix

• Identity matrix

• Symmetric matrix

• Skew symmetric matrix

Q.1 | Q 7

Choose the correct alternative:

If A and B are two square matrices of order 3, then (AB)T = ______

• BTAT

• ATBT

• (BA)T

• I

Q.1 | Q 8

Choose the correct alternative:

If A = [(1, 2),(2, -1)], then adj (A) = ______

• [(1, -2),(2, -1)]

• [(-1, 2),(2, 1)]

• [(-1, -2),(-2, 1)]

• [(1, 2),(-2, -1)]

Q.1 | Q 9

Choose the correct alternative:

If A is a non singular matrix of order 3, then |adj (A)| =  ______

• |A|2

• |A|3

• 0

• 1

Q.1 | Q 10

Choose the correct alternative:

If A2 + 5A + 3I = 0, |A| ≠ 0, then A–1 = ______

• (-1)/3 ("A" + 5"I")

• (-1)/5 ("A" + 3"I")

• (A + 15I)

• (-1)/3 ("I" + 5"A")

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.2

#### [1 mark]

Q.2 | Q 1

State whether the following statement is True or False:

If A is non singular, then |A| = 0

• True

• False

Q.2 | Q 2

State whether the following statement is True or False:

Inverse of [(2, 0),(0, 3)] is [(1/2, 0),(0, 1/3)]

• True

• False

Q.2 | Q 3

State whether the following statement is True or False:

If [(3, 0),(0, 2)][(x),(y)] = [(3),(2)], then x = 1 and y = – 1

• True

• False

Q.2 | Q 4

State whether the following statement is True or False:

Every square matrix of order n can be expressed as sum of symmetric and skew symmetric matrix

• True

• False

Q.2 | Q 5

State whether the following statement is True or False:

If A = [(1, 2, -5),(2, -3, 4),(-5, 4, 9)], then AT = A

• True

• False

Q.2 | Q 6

State whether the following statement is True or False:

Matrix [("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")] is a singular

• True

• False

Q.2 | Q 7

State whether the following statement is True or False:

[(2, 0, 0),(3, -1, 0),(-7, 3, 1)] is a skew symmetric matrix

• True

• False

Q.2 | Q 8

State whether the following statement is True or False:

If O(A) = m × n and O(B) = n × p with m ≠ p, then BA exists but AB does not exist.

• True

• False

Q.2 | Q 9

State whether the following statement is True or False:

After applying elementary transformation R1 – 3R2 on matrix [(3, -2),(1, 4)] we get [(0, -12),(1, 4)]

• True

• False

Q.2 | Q 10

State whether the following statement is True or False:

If A and B are two square matrices such that AB = BA, then (A – B)2 = A2 – 2AB + B2

• True

• False

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.3

#### Fill in the blanks [1 Mark]

Q.3 | Q 1

If A = [aij]2×2, where aij = i – j, then A = ______

Q.3 | Q 2

In a Skew symmetric matrix, all diagonal elements are ______

Q.3 | Q 3

If A = [(1, -2),(5, 3)], B = [(1, -3),(4, -7)], then A – 3B = ______

Q.3 | Q 4

If A = [(4, 3, 2),(-1, 2, 0)], B = [(1, 2),(-1, 0),(1, -2)], then |AB| = ______

Q.3 | Q 5

The value of Cofactor of element a21 in matrix A = [(1, 2),(5, -8)] is ______

Q.3 | Q 6

The value of Minor of element b22 in matrix B = [(2, -2),(4, 5)] is ______

Q.3 | Q 7

If matrix form of given equations 3x – y = 1 and y + 4x = 6 is AX = B, then A = ______

Q.3 | Q 8

The suitable elementary row transformation which will reduce the matrix [(1, 0),(2, 1)] into identity matrix is ______

Q.3 | Q 9

If A = [(2, 0, 0),(0, 1, 0),(0, 0, 1)], then |adj (A)| = ______

Q.3 | Q 10

If A is a square matrix of order 2 such that A(adj A) = [(7, 0),(0, 7)], then |A| = ______

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.4

#### Solve the following [3 Marks]

Q.4 | Q 1

If A = [(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")] and (A + B)2  A2 + B2, find the values of a and b.

Q.4 | Q 2

Find matrices A and B, if 2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]

Q.4 | Q 3

If A = [(3, 1),(-1, 2)], then prove that A2 – 5A + 7I = O, where I is unit matrix of order 2

Q.4 | Q 4

Find k, if A = [(3, -2),(4, -2)] and A2 = kA – 2I, where I is identity matrix of order 2

Q.4 | Q 5

If A = [(2, 1),(0, 3),(1, -1)] and B = [(0, 3, 5),(1, -7, 2)], then verify (BA)T = ATBT

Q.4 | Q 6

Find the inverse of matrix B = [(3,1, 5),(2, 7, 8),(1, 2, 5)] by using adjoint method

Q.4 | Q 7

If A = [(3, 1),(1, 5)] nd B = [(1, 2),(5, -2)], then verify |AB| = |A||B|

Q.4 | Q 8

If A = [(1, 3, 3),(3, 1, 3),(3, 3, 1)], then show that A2 – 5A is a scalar matrix

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.5

#### [4 Marks]

Q.5 | Q 1

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

Q.5 | Q 2

Solve the following equations by method of inversion :

4x – 3y – 2 = 0, 3x – 4 + 6 = 0

Q.5 | Q 3

Solve the following equations by method of reduction :

x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3

Q.5 | Q 4

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

Q.5 | Q 5

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.

Q.6

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.6

#### Activities

Q.6 | Q 1

Complete the following activity to verify A. adj (A) = det (A) I.

Given A = [(2, 0, -1),(5, 1, 0),(0, 1, 3)] then

|A| = 2(____) – 0(____) + ( ) (____)

= 6 – 0 – 5

= ______ ≠ 0

Cofactors of all elements of matrix A are

A11 = (-1)^2 |("( )", "( )"),("( )", "( )")| = (______),

A12 = (-1)^3 |(5, "( )"),("( )", 3)| = – 15,

A13 = (-1)^4 |(5, "( )"),("( )", 1)| = 5,

A21 = _______, A22 = _______, A23 = _______,

A31 = (-1)^4 |("( )", "( )"),("( )", "( )")| = (______),

A32 = (-1)^5 |(2, "( )"),("( )", 0)| = (  ),

A33 = (-1)^6 |(2, "( )"),("( )", 1)| = 2,.

Cofactors of matrix A = [(3, "____", "____"),("____", "____",-2),(1, "____", "____")]

adj (A) = [("____", "____", "____"),("____", "____","____"),("____","____","____")]

A.adj (A) = [(2, 0, -1),(5, 1, 0),(0, 1, 3)] [("( )", -1, 1), (-15, "( )", -5),("( )", -2, "( )")] = [(1, 0, "( )"),("( )", "( )", "( )"),(0, "( )", "( )")] = |A|I

Q.6 | Q 2

Complete the following activity to find inverse of matrix using elementary column transformations and hence verify.

[(2, 0, -1),(5, 1, 0),(0, 1, 3)] B−1 = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]

C1 → C1 + C3

[("( )", 0, -1),("( )", 1, 0),("( )", 1, 3)] B−1 = [("( )", 0, 0),("( )", 1, 0),("( )", 0, 1)]

C3 → C3 + C1

[(1, 0, 0),("( )", 1, "( )"),(3, 1, "( )")] B−1 = [(1, 0, "( )"),(0, 1, 0),("( )", 0, "( )")]

C1 → C1 – 5C2, C3 → C3 – 5C2

[(1, "( )", 0),(0, 1, 0),("( )", 1, "( )")] B−1 = [(1, 0, "( )"),("( )", 1, -5),(1, "( )", 2)]

C1 → C1 – 2C3, C2 → C2 – C

[(1, 0, 0),(0, 1, 0),(0, 0, 1)] B−1 = [(3, -1, "( )"),("( )", 6, -5),(5, "( )", "( )")]

B−1 =  [("( )", "( )", "( )"),("( )", "( )", "( )"),("( )", "( )", "( )")]

[(2, "( )", -1),("( )", 1, 0),(0, 1, "( )")] [(3, "( )", "( )"),("( )", 6, "( )"),("( )", -2, "( )")] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]

Q.6 | Q 3

Complete the following activity.

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]

R1 ↔ R2

[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]

R2 – 4R1, R3 – 6R1

[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]

(-1)/5 "R"_2, (-1)/5 "R",

[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]

R3 – 2R2

[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]

By pre multiplying we get,

x + 2y + ( )z = ( )    .....(i)

y + 2z = 24    ......(ii)

–z = ( )      ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

## Chapter 1: Matrices

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## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Matrices

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 (Matrices) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 Matrices are Determinant of a Matrix, Types of Matrices, Algebra of Matrices, Properties of Matrices, Elementary Transformations, Inverse of Matrix, Application of Matrices.

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