SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.2 - Matrices [Latest edition]

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

Choose the correct alternative:

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

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.

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

For any square matrix B, matrix B + B^T is ______

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

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

State whether the following statement is True or False:

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

State whether the following statement is True or False:

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

State whether the following statement is True or False:

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

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

State whether the following statement is True or False:

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

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

State whether the following statement is True or False:

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

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.

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)]`

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 

If A = [a_ij]_2×2, where a_ij = i – j, then A = ______

In a Skew symmetric matrix, all diagonal elements are ______

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

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

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

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

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

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

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

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

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

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)]` 

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

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

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

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

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

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

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

Solve the following equations by method of inversion :

4x – 3y – 2 = 0, 3x – 4y + 6 = 0

Solve the following equations by method of reduction :

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

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.

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.

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

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 – C₃ 

`[(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)]`

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"_3`

`[("( )", 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.