Balbharati solutions for Mathematics and Statistics 1 (Commerce) [English] 12 Standard HSC Maharashtra State Board chapter 2 - Matrices [Latest edition]

Construct  a matrix A = `[a_("ij")]_(3 xx 2)` whose element a_ij is given by

a_ij = `((i - j)^2)/(5 - i)`

Construct a matrix A = `[a_("ij")]_(3 xx 2)` whose element a_ij is given by

a_ij = i – 3j

Construct a matrix A = `[a_("ij")]_(3 xx 2)` whose element a_ij is given by

a_ij = `(i + j)^3/(5)`

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

`[(5),(4),(-3)]`

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

`[9   sqrt(2)   -3]`

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

`[(6, 0),(0, 6)]`

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

`[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]`

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`

Classify each of the following matrices as a row, a column, a square, a diagonal, a scalar, a unit, an upper traingular, a lower triangular matrix.

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

Which of the following matrices are singular or non singular?

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

Which of the following matrices are singular or non singular?

`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`

Which of the following matrices are singular or non singular?

`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`

Which of the following matrices are singular or non singular?

`[(7, 5),(-4, 7)]`

Find K if the following matrices are singular.

`[(7, 3),(-2, "K")]`

Find K if the following matrices are singular.

`[(4, 3, 1),(7, "K", 1),(10, 9, 1)]`

Find K if the following matrices are singular.

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

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2),(0, 3)] "and C" = [(4, 3),(-1, 4),(-2, 1)]`, Show that A + B = B + A

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(-1, 2),(2, 2),(0, 3)] "and C" = [(4, 3),(-1, 4),(-2, 1)]`, Show that (A + B) + C = A + (B + C)

If A = `[(1, -2),(5, 3)], "B" = [(1, -3),(4, -7)]` , then find the matrix A − 2B + 6I, where I is the unit matrix of order 2.

If A = `[(1, 2, -3),(-3, 7, -8),(0, -6, 1)], "B" = [(9, -1, 2),(-4, 2, 5),(4, 0, -3)]` then find the matrix C such that A + B + C is a zero matrix.

If A = `[(1, -2),(3, -5),(-6, 0)],"B" = [(-1, -2),(4, 2),(1, 5)] "and C" = [(2, 4),(-1, -4),(-3, 6)]`, find the matrix X such that 3A – 4B + 5X = C.

If A = `[(5, 1, -4),(3, 2, 0)]`, find (A^T)^T.

If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, find (A^T)^T.

Find a, b, c, if `[(1, 3/5, "a"),("b", -5, -7),(-4, "c", 0)]` is a symmetric matrix.

Find x, y, z if `[(0, -5i, x),(y, 0, z),(3/2, - sqrt(2), 0)]` is a skew symmetric matrix.

For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.

`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`

For each of the following matrices, find its transpose and state whether it is symmetric, skew-symmetric, or neither.

`[(2, 5, 1),(-5, 4, 6),(-1, -6, 3)]`

For each of the following matrices, find its transpose and state whether it is symmetric, skew- symmetric or neither.

`[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]`

Construct the matrix A = [a_ij]_3×3 where a_ij = i − j. State whether A is symmetric or skew-symmetric.

Solve the following equations for X and Y, if 3X − Y = `[(1, -1),(-1, 1)]`  and X – 3Y = `[(0, -1),(0, -1)]`.

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

Find x and y, if `[(2x + y, -1, 1),(3, 4y, 4)] [(-1,  6, 4),(3, 0, 3)] = [(3, 5, 5),(6, 18, 7)]`.

If `[(2"a" + "b", 3"a" - "b"),("c" + 2"d", 2"c" - "d")] = [(2, 3),(4, -1)]`, find a, b, c and d.

There are two book shops own by Suresh and Ganesh. Their sales (in Rupees) for books in three subject - Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B.

July sales (in Rupees), Physics Chemistry Mathematics

A = `[(5600, 6750, 8500),(6650, 7055, 8905)][("Suresh"), ("Ganesh")]`

August Sales (in Rupees) Physics Chemistry Mathematics

B = `[(6650, 7055, 8905),(7000, 7500, 10200)][("Suresh"), ("Ganesh")]`

Find the increase in sales in Rupees from July to August 2017.

There are two book shops own by Suresh and Ganesh. Their sales ( in Rupees) for books in three subject - Physics, Chemistry and Mathematics for two months, July and August 2017 are given by two matrices A and B. July sales ( in Rupees) :
Physics Chemistry Mathematics
A = `[(5600, 6750, 8500),(6650, 7055, 8905)][("Suresh"), ("Ganesh")]`
August Sales (in Rupees :
B = `[(6650, 7055, 8905),(7000, 7500, 10200)][("Suresh"), ("Ganesh")]`
If both book shops get 10% profit in the month of August 2017, find the profit for each book seller in each subject in that month.

Evaluate : `[(3),(2),(1)][2  -4   3]`

Evaluate : `[2 - 1   3][(4),(3),(1)]`

If A = `[(-1, 1, 1),(2, 3, 0),(1, -3, 1)],"B" = [(2, 1, 4),(3, 0, 2),(1, 2, 1)]`, state whether AB = BA? Justify your answer.

Show that AB = BA, where A = `[(-2, 3, -1),(-1, 2, -1),(-6, 9, -4)],"B" = [(1, 3, -1),(2, 2, -1),(3, 0, -1)]`.

Verify A(BC) = (AB)C, if A = `[(1, 0, 1),(2, 3, 0),(0, 4, 5)], "B" = [(2, -2),(-1, 1),(0, 3)] and "C" = [(3,2,-1), (2,0,-2)]`

Verify that A(B + C) = AB + AC, if A = `[(4, -2),(2, 3)], "B" = [(-1, 1),(3, -2)] " and C" = [(4 ,1),(2, -1)]`.

If  A = `[(4, 3, 2),(-1, 2, 0)],"B" = [(1, 2),(-1, 0),(1, -2)]` show that matrix AB is non singular.

If A + I = `[(1, 2, 0),(5, 4, 2),(0, 7, -3)]`, find the product (A + I)(A − I).

If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, show that A² – 4A is a scalar matrix.

If A = `[(1, 0),(-1, 7)]`, find k, so that A² – 8A – kI = O, where I is a 2 × 2 unit and O is null matrix of order 2.

If A = `[(3, 1),(-1, 2)]`, prove that A² – 5A + 7I = 0, where I is a 2 x 2 unit matrix.

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 k, if A = `[(3, -2),(4, -2)]` and A² = kA – 2I.

Find x and y, if `{4[(2, -1, 3),(1, 0, 2)] - [(3, -3, 4),(2, 1, 1)]}[(2),(-1),(1)] = [(x),(y)]`

Find x, y, x, if `{3[(2, 0),(0, 2),(2, 2)] -4[(1, 1),(-1, 2),(3, 1)]} [(1),(2)] = [(x - 3),(y - 1),(2z)]`.

Jay and Ram are two friends. Jay wants to buy 4 pens and 8 notebooks, Ram wants to buy 5 pens and 12 notebooks. The price of one pen and one notebook was ₹ 6 and ₹ 10 respectively. Using matrix multiplication, find the amount each one of them requires for buying the pens and notebooks.

Find A^T,  if A = `[(1, 3),(-4, 5)]`

Find A^T, if A = `[(2, -6, 1),(-4, 0, 5)]`

If [a_ij]_3×3, where a_ij = 2(i – j), find A and A^T. State whether A and A^T both are symmetric or skew-symmetric matrices?

If A = `[(5, -3),(4, -3),(-2, 1)]`, prove that (A^T)^T = A.

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

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(2, 1),(4, -1),(-3, 3)], "C" = [(1, 2),(-1, 4),(-2, 3)]`, then show that (A + B)^T = A^T + B^T.

If A = `[(2, -3),(5, -4),(-6, 1)], "B" = [(2, 1),(4, -1),(-3, 3)], "C" = [(1, 2),(-1, 4),(-2, 3)]`, then show that (A – C)^T = A^T – C^T.

If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find C^T, such that 3A – 2B + C = I, whre I is e unit matrix of order 2.

If A = `[(7, 3, 0),(0, 4, -2)], "B" = [(0, -2, 3),(2, 1, -4)]`, then find A^T + 4B^T.

If A = `[(7, 3, 0),(0, 4, -2)], "B" = [(0, -2, 3),(2, 1, -4)]`, then find 5A^T – 5B^T.

If A = `[(1, 0, 1),(3, 1, 2)], "B" = [(2, 1, -4),(3, 5, -2)] "and"  "C" = [(0, 2, 3),(-1, -1, 0)]`, verify that (A + 2B + 3C)^T = A^T + 2B^T + C^T.

If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + B^T)^T = A^T + B.

Prove that A + A^T is a symmetric and A – A^T is a skew symmetric matrix, where A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`

Prove that A + A^T is a symmetric and A – A^T is a skew symmetric matrix, where A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`

Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix `[(4, -2),(3, -5)]`.

Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`.

If A = `[(2, -1),(3, -2),(4, 1)] "and B" = [(0, 3, -4),(2, -1, 1)]`, verify that (AB)^T = B^TA^T.

If A = `[(2, -1),(3, -2),(4, 1)] "and B" = [(0, 3, -4),(2, -1, 1)]`, verify that (BA)^T = A^TB^T.

Apply the given elementary transformation on each of the following matrices `[(3, -4),(2, 2)]`, R₁ ↔ R₂.

Apply the given elementary transformation on each of the following matrices `[(2, 4),(1, -5)]`, C₁ ↔ C₂.

Apply the given elementary transformation on each of the following matrices `[(3, 1, -1),(1, 3, 1),(-1, 1, 3)]`, 3R₂ and C₂ ↔ C₂ – 4C₁.

Transform `[(1, -1, 2),(2, 1, 3),(3, 2, 4)]` into an upper traingular matrix by suitable row transformations.

Find the cofactor matrix, of the following matrices : `[(1, 2),(5, -8)]`

Find the cofactor matrix, of the following matrices: `[(5, 8, 7),(-1, -2, 1),(-2, 1, 1)]`

Find the adjoint of the following matrices : `[(2, -3),(3, 5)]`

Find the adjoint of the following matrices : `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`

Find the inverse of the following matrices by the adjoint method `[(3, -1),(2, -1)]`.

Find the inverse of the following matrices by the adjoint method `[(2, -2),(4, 5)]`.

Find the inverse of the following matrices by the adjoint method `[(1, 2, 3),(0, 2, 4),(0, 0, 5)]`.

Find the inverse of the following matrices by transformation method: `[(1, 2),(2, -1)]`

Find the inverse of the following matrices by transformation method:

`[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`

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

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

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.

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

Solve the following equations by method of inversion.
x + 2y = 2, 2x + 3y = 3

Solve the following equations by method of inversion.
2x + y = 5, 3x + 5y = – 3

Solve the following equation by the method of inversion.

2x – y + z = 1,
x + 2y + 3z = 8,
3x + y – 4z = 1

Solve the following equations by method of inversion.
x + y + z = 1, x – y + z = 2 and x + y – z = 3

Express the following equations in matrix form and solve them by method of reduction.
x + 3y  = 2, 3x + 5y = 4

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

3x – y = 1, 4x + y = 6

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 method of reduction.

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

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.

The sum of the cost of one Economic book, one Co-operation book and one account book is ₹ 420. The total cost of an Economic book, 2 Co-operation books and an Account book is ₹ 480. Also the total cost of an Economic book, 3 Co-operation books and 2 Account books is ₹ 600. Find the cost of each book using matrix method.

Choose the correct alternative.

If AX = B, where A = `[(-1, 2),(2, -1)], "B" = [(1),(1)]`, then X = _______

Choose the correct alternative.

The matrix `[(8, 0, 0),(0, 8, 0),(0, 0, 8)]` is _______

Choose the correct alternative.

The matrix `[(0, 0, 0),(0, 0, 0)]` is _______

Choose the correct alternative.

If A = `[("a", 0, 0),(0, "a", 0),(0, 0,"a")]`, then |adj.A| = _______

Adjoint of `[(2, -3),(4, -6)]` is _______

Choose the correct alternative.

If A = diag [d₁, d₂, d₃,...,d_n], where di ≠ 0, for i = 1, 2, 3,...,n, then A^–1 = _______

Choose the correct alternative.

If A² + mA + nI = O and n ≠ 0, |A| ≠ 0, then A^–1 = _______

Choose the correct alternative.

If a 3 x 3 matrix B has it inverse equal to B, thenB² = _______

Choose the correct alternative.

If A = `[(α, 4),(4, α)]` and |A³| = 729, then α = ______.

Choose the correct alternative.

If A and B are square matrices of order n × n such that A² – B² = (A – B)(A + B), then which of the following will be always true?

Choose the correct alternative.

If A = `[(2, 5),(1, 3)]`, then A^–1 = _______

Choose the correct alternative.

If A is a 2 x 2 matrix such that A(adj. A) = `[(5, 0),(0, 5)]`, then |A| = _______

If A is a no singular matrix, then det (A^–1) = _______

If A = `[(1, 2),(-3, -1)], "B" = [(-1, 0),(1, 5)]`, then AB = 

If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = _______

Fill in the blank:

A = `[(3),(1)]` is ........................ matrix.

Fill in the blank :

Order of matrix `[(2, 1, 1),(5, 1, 8)]` is _______

Fill in the blank :

If A = `[(4, x),(6, 3)]` is a singular matrix, then x is _______

Fill in the blank :

Matrix B = `[(0, 3, 1),(-3, 0, -4),("p", 4, 0)]` is skew symmetric, then the value of p is _______

Fill in the blank :

If A = [a_ij]_2x3 and B = [b_ij]_mx1 and AB is defined, then m = _______

Fill in the blank :

If A = `[(3, -5),(2, 5)]`, then co-factor of a₁₂ is _______

Fill in the blank :

If A = [a_ij]_mxm is a non-singular matrix, then A^–1 = `(1)/(......)` adj(A).

Fill in the blank :

(A^T)^T = _______

Fill in the blank :

If A = `[(2, 1),(1, 1)] "and" "A"^-1 = [(1, 1),(x, 2)]`, then x = _______

Fill in the blank :

If a₁x + b₁y = c₁ and a₂x + b₂y = c₂, then matrix form is `[(......, ......),(......, ......)] = [(x),(y)] = [(......),(......)]`

State whether the following is True or False :

Single element matrix is row as well as column matrix.

State whether the following is True or False :

Every scalar matrix is unit matrix.

State whether the following is True or False :

A = `[(4, 5),(6, 1)]` is no singular matrix.

State whether the following is True or False :

If A is symmetric, then A = –A^T.

State whether the following is True or False :

If AB and BA both exist, then AB = BA.

State whether the following is True or False :

If A and B are square matrices of same order, then (A + B)² = A² + 2AB + B².

State whether the following is True or False :

If A and B are conformable for the product AB, then (AB)^T = A^TB^T.

State whether the following is True or False :

Singleton matrix is only row matrix.

State whether the following is True or False :

A = `[(2, 1),(10, 5)]` is invertible matrix.

State whether the following is True or False :

A(adj. A) = |A| I, where I is the unit matrix.

Solve the following :

Find k, if `[(7, 3),(5, "k")]` is a singular matrix.

Solve the following :

Find x, y, z if `[(2, x, 5),(3, 1, z),(y, 5, 8)]` is a symmetric matrix.

Solve the following :

If A = `[(1, 5),(7, 8),(9, 5)], "B" = [(2, 4),(1, 5),(-8, 6)] "C" = [(-2, 3),(1, -5),(7, 8)]` then show that (A + B) + C = A + (B + C).

Solve the following :

If A = `[(2, 5),(3, 7)], "B" = 4[(1, 7),(-3, 0)]`, find matrix A – 4B + 7I, where I is the unit matrix of order 2.

Solve the following :

If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (A + 2B^T)^T = A^T + 2B.

Solve the following :

If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (3A – 5B^T)^T = 3A^T – 5B.

Solve the following :

If A = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)],"B" = [(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`, then show that AB and BA are bothh singular martices.

Solve the following :

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

Solve the following :

If A = `[(2, -1),(-1, 2)]`, then show that A² – 4A + 3I = 0.

Solve the following :

If A = `[(-3, 2),(2, 4)], "B" = [(1, "a"), ("b", 0)]` and (A + B) (A – B) = A² – B², find a and b.

Solve the following :

if A = `[(1, 2),(-1, 3)]`, then find A³.

Find x, y, z, if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, - 2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]`

Solve the following :

If A = `[(2, -4),(3, -2),(0, 1)], "B" = [(1, -1, 2),(-2, 1, 0)]`, then show that (AB)^T = B^TA^T.

Solve the following :

If A = `[(1, 0, 0),(2, 1, 0),(3, 3, 1)]`, the reduce it to unit matrix by using row transformations.

Solve the following :

Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (in Rupees) of these crops by both the farmers for the month of April and May 2016 is given below,

Find : The total sale in rupees for two months of each farmer for each crop.

Solve the following :

Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (in Rupees) of these crops by both the farmers for the month of April and May 2016 is given below,

Find : the increase in sale from April to May for every crop of each farmer.

Check whether the following matrices are invertible or not:

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

Check whether the following matrices are invertible or not:

`[(1, 1),(1, 1)]`

Check whether the following matrices are invertible or not:

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

Check whether the following matrices are invertible or not:

`[(1, 2, 3),(2, 4, 5),(2, 4, 6)]`

Find inverse of the following matrices (if they exist) by elementary transformations :

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

Find inverse of the following matrices (if they exist) by elementary transformations :

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

Find inverse of the following matrices (if they exist) by elementary transformations :

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

Find inverse of the following matrices (if they exist) by elementary transformations :

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

Find the inverse of `[(3, 1, 5),(2, 7, 8),(1, 2, 5)]` by adjoint method.

Solve the following equations by method of inversion : x + y – z = 2, x – 2y + z = 3 and 2x – y – 3z = – 1

Solve the following equations by method of inversion : x – y + z = 4, 2x + y – 3z = 0 , x + y + z = 2

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

Solve the following equations by method of reduction :

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

Solve the following equations by the reduction method.

2x + y = 5, 3x + 5y = – 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.