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

#### Chapters ## Chapter 1: Matrices

Q.1Q.2Q.3Q.4Q.5Q.6
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)] = ______

• 

• 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

Q.1Q.2Q.3Q.4Q.5Q.6 ## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Matrices

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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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