If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

If and then find

a₁₁ b₁₁ + a₂₂ b₂₂

Let A be a matrix of order 3 × 4. If R₁ denotes the first row of A and C₂ denotes its second column, then determine the orders of matrices R₁ and C₂.

Construct a 2 ×3 matrix A = [a_jj] whose elements a_jj are given by :

a_ij = i × j

Construct a 2 ×3 matrix A = [a_jj] whose elements a_jj are given by :

a_ij = 2i – j

Construct a 2 ×3 matrix A = [a_jj] whose elements a_jj are given by :

a_ij = i + j

Construct a 2 ×3 matrix A = [a_jj] whose elements a_jj are given by :

a_ij =

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 2 × 2 matrix A = [a_jj] whose elements a_jj are given by :

a_ij = e^2ix sin xj

Construct a 3×4 matrix A = [a_jj] whose elements a_jj are given by :

a_jj = i + j

Construct a 3×4 matrix A = [a_jj] whose elements a_jj are given by :

a_jj = i – j

Construct a 3×4 matrix A = [a_jj] whose elements a_jj are given by :

a_jj = 2i

Construct a 3×4 matrix A = [a_jj] whose elements a_jj are given by :

a_jj = j

Construct a 3×4 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 4 × 3 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 4 × 3 matrix A = [a_jj] whose elements a_jj are given by :

Construct a 4 × 3 matrix A = [a_jj] whose elements a_jj are given by :

a_jj = i

Find x, y, a and b if

Find x, y, a and b if

Find the values of a, b, c and d from the following equations:

Find x, y and z so that A = B, where

If find x, y, z, ω.

If find x, y, z,ω.

If

Obtain the values of a, b, c, x, y and z.

If find the value of (x + y).

If then find the values of x, y, z and ω.

Give an example of

a row matrix which is also a column matrix

Give an example of

a diagonal matrix which is not scalar

Give an example of

a triangular matrix.

The sales figure of two car dealers during January 2013 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 standard cars. Total sales over the 2 month period of January – February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars. In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars. Write 2 x 3 matrices summarizing sales data for January and 2 – month period for each dealer.

For what value of x and y are the following matrices equal?

Find the values of x and y if

Find the values of a and b if A = B, where

Compute the following sums:

Compute the following sums:

Let and Find each of the following :

i. 2A – 3B

ii. B – 4C

iii. 3A – C

iv. 3A – 2B + 3C

If find

i. A + B and B + C

ii. 2B + 3A and 3C – 4B.

Let and Compute 2A – 3B + 4C.

If A = diag (2, -5, 9), B = diag (1, 1, -4) and C = diag (-6, 3, 4), find

i. A – 2B

ii. B + C – 2A

iii. 2A + 3B – 5C

Given the matrices

and

Verify that (A + B) + C = A + (B + C).

Find matrices X and Y, if X + Y = and

Find X, if and

Find matrices X and Y, if and

If and find X and Y.

Find matrix A, if

If find matrix C such that 5A + 3B + 2C is a null matrix.

If find matrix X such that 2A + 3X = 5B.

If and find the matrix C such that A + B + C is zero matrix.

Find x, y satisfying the matrix equations

Find x, y satisfying the matrix equations

If find x and y.

Find the value of λ, a non-zero scalar, if .

Find a matrix X such that 2A + B + X = O, where

If and then find the matrix X of order 3 x 2 such that 2A + 3X = 5B.

Find x, y, z and t, if

Find x, y, z and t, if

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.

The monthly incomes of Aryan and Babban are in the ration 3: 4 and their monthly expenditures are in the ratio 5: 7. If each saves 15000 per month, find their monthly incomes using the matrix method. This problem reflects which value?

Compute the indicated products:

Compute the indicated products:

Compute the indicated products:

Show that AB ≠ BA in each of the following cases:

and

Show that AB ≠ BA in each of the following cases:

and

Show that AB ≠ BA in each of the following cases:

and

Compute the products AB and BA whichever exists in each of the following cases:

and

Compute the products AB and BA whichever exists in each of the following cases:

and

Compute the products AB and BA whichever exists in each of the following cases:

and

Compute the products AB and BA whichever exists in each of the following cases:

Show that AB ≠ BA in each of the following cases:

and

Show that AB ≠ BA in each of the following cases:

and

Evaluate the following:

Evaluate the following:

Evaluate the following:

If and then show that A² = B² = C² = I₂.

If and find 3A² – 2B + I.

If prove that (A – 2I) (A – 3I) = O.

If Show that and

If show that A² = O

If find A²

If and show that AB = BA = O_{3 × 3}.

If and show that AB = BA = O_{3 × 3}.

If and show that AB = A and BA = B.

Let and compute A² – B².

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC).

and

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC).

and

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A(B + C) = AB + AC.

and

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A(B + C) = AB + AC.

and

If and verify that A(B – C) = AB – AC

Compute the elements a₄₃ and a₂₂ of the matrix:

If and I is the identity matrix of order 3, show that A³ = pl + qA + rA².

If ω is a complex cube root of unity, show that

If show that A² = A.

If show that A² = I₃

If find x.

If find x.

If find x

If find x.

If and then prove that A² – A + 2I = 0.

If and then find λ so that A² = 5A + λI.

If show that A² – 5A + 7I₂ = 0.

If show that A² – 2A + 3I₂ = 0.

Show that the matrix satisfies the equation A³ – 4A² + A = 0.

Show that the matrix is a root of the equation A² – 12A – I = 0.

If find A² – 5A – 14I.

If show that A² –5A + 7I = 0. Use this to find A⁴.

If find k such that A² = kA – 2I₂.

If find k such that A² – 8A + kI – 0.

If and f(x) = x² – 2x – 3, show that f(A) = 0.

If and then find λ, μ so that A² = λ A + μ I

Find the value of x for which the matrix product equal to an identity matrix.

Solve the matrix equations:

Solve the matrix equations:

Solve the matrix equations:

Solve the matrix equations:

If compute A² – 4A + 3I₃.

If f(x) = x² – 2x, find f(A), where

If f(x) = x³ + 4x² – x, find f(A), where

If then show that A is a root of the polynomial f(x) = x³ – 6x² + 7x + 2.

If then prove that A² – 4A – 5I = 0.

If show that A² – 7A + 10I₃ = 0.

Without using the concept of the inverse of a matrix, find the matrix such that

Find the matrix A such that

Find the matrix A such that

Find the matrix A such that

Find the matrix A such that

Find the matrix A such that

Find the matrix A such that

Find a 2 × 2 matrix A such that

If find A¹⁶.

If and x² = – 1, then show that (A + B)² = A² + B².

If then verify that A² + A = (A + I), where I is the identity matrix.

If then find A² – 5A – 14I. Hence, obtain A³.

If then show that P(x) P(y) = P(x + y) = P(y) P(x).

If and prove that

If find A² – 5A + 4I and hence find a matrix X such that A² – 5A + 4I + X = O.

If prove that for all positive integers n.

If prove that for every positive integer n.

If then prove by principle of mathematical induction that for all n ∈ N.

If prove that for all

If then use the principle of mathematical induction to show that for every positive integer n.

If B, C are n rowed matrices and if A = B + C, BC = CB, C² = O, then show that for every n ϵ N, Aⁿ⁺¹ = Bⁿ(B + (n + 1)C).

If A = diag(a b c), show that Aⁿ = diag(aⁿ bⁿ cⁿ) for all positive integers n.

If A is a square matrix, using mathematical induction prove that (A^T)ⁿ = (Aⁿ)^T for all n ϵ N.

A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal?

Give examples of matrices

A and B such that AB ≠ BA.

Give examples of matrices

A and B such that AB = O but A ≠ O, B ≠ O.

Give examples of matrices

A and B such that AB = O but BA ≠ O.

Give examples of matrices

A, B and C such that AB = AC but B ≠ C, A ≠ O.

Let A and B be square matrices of the same order. Does (A + B)² = A² + 2AB + B² hold? If not, why?

If A and B are square matrices of the same order, explain, why in general

(i) (A + B)² ≠ A² + 2AB + B²

(ii) (A – B)² ≠ A² – 2AB + B²

(iii) (A + B)(A – B) = A² – B²

Let A and B be square matrices of the order 3 × 3. Is (AB)² = A²B²? Give reasons.

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

Let and

Verify that AB = AC though B ≠ C, A ≠ 0.

Three shopkeepers, A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A notebook costs 40 paise, a pen costs Rs 1.25 and a pencil costs 35 paise. Use matrix multiplication to calculate each individual’s bill.

The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs 8.30, Rs 3.45 and Rs 4.50 each respectively. Find the total amount the store will receive from selling all the items.

In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways; telephone, house calls and letters. The cost per contact (in paise) is given matrix A as

The number of contacts of each type made in two cities X and Y is given in matrix B as

Find the total amount spent by the group in the two cities X and Y.

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year and the second bond pays 7% per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds if the trust fund must obtain an annual total interest of (i) Rs 1800 and (ii) Rs 2000.

75. To promote making of toilets for women, an organization tried to generate awareness through (i) house calls (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below :

(i) ₹ 50

(ii) ₹ 20

(iii) ₹ 40

The number of attempts made in three villages X, Y and Z are given below :

Find the total cost incurred by the organization for three villages separately, using matrices.

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommended daily amount of calories is 2400 for men, 1990 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using a matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create among people about the planned diet from this question?

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways – telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as

The number of contacts of each type made in two cities X and Y is given in matrix B as

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote – party’s promotional activity or their social activities?

The monthly incomes of Aryan and Babbar are in the ratio 3:4 and their monthly expenditures are in the ratio 5:7. If each saves Rs 15000 per month, find their monthly incomes using matrix method. This problem reflects which value?

A trust invested some money in two types of bonds. The first bond pays 10% interest and the second bond pays 12%. The trust received Rs 2800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust.

Let and verify that

(A + B)^T = A^T + B^T

Let and verify that

(A – B)^T = A^T - B^T

Let and verify that

(AB)^T = B^TA^T

If and B = [1 0 4], verify that (AB)^T = B^TA^T.

Let and Find A^T, B^T and verify that

(A + B)^T = A^T + B^T

Let and Find A^T, B^T and verify that

(AB)^T = B^T A^T

Let and Find A^T, B^T and verify that

(2A)^T = 2A^T

If B= [1 3 -6], verify that (AB)^T = B^TA^T.

If find (AB)^T .

For two matrices A and B, verify that (AB)^T = B^TA^T.

For the matrices, A and B, verify that (AB)^T = B^TA^T, where

If and find A^T – B^T.

If then verify that A^TA = I₂.

If verify that A^TA = I₂.

If denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^T = I, where

If show that A – A^T is a skew-symmetric matrix.

If the matrix is a symmetric matrix, find x, y, z and t.

Let Find matrices X and Y such that X + Y = A, where X is a symmetric and Y is a skew symmetric matrix.

Express the matrix as the sum of a symmetric and a skew-symmetric matrix.

Define a symmetric matrix. Prove that for is a symmetric matrix where A^T is the transpose of A.

Express the matrix as the sum of a symmetric and a skew-symmetric matrix.

Express the matrix as the sum of a symmetric and skew-symmetric matrix and verify your result.

If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.

If and , write AB.

If , write AA^T.

Give an example of two non-zero 2 × 2 matrices A and B such that AB = O.

If , find A + A^T.

If , write A².

If , find x satisfying when A + A^T = I.

If , find AA^T.

If , where I is 2 × 2 unit matrix. Find x and y.

If , satisfies the matrix equation A² = kA, write the value of k.

If satisfies A⁴ = λA, then write the value of λ.

If , find A².

If , find A³.

If , find A⁴.

If [x 2], find x.

If A = [a_ij] is a 2 × 2 matrix such that a_ij = i + 2j, write A.

Write matrix A satisfying .

If A = [a_ij] is a square matrix such that a_ij = i² – j², then write whether A is symmetric or skew-symmetric.

For any square matrix write whether AA^T is symmetric or skew-symmetric.

If A = [a_ij] is a skew-symmetric matrix, then write the value of .

If A = [a_ij] is a skew-symmetric matrix, then write the value of .

If A and B are symmetric matrices, then write the condition for which AB is also symmetric.

If B is a skew-symmetric matrix, write whether the matrix AB A^T is symmetric or skew-symmetric.

If B is a symmetric matrix, write whether the matrix AB A^T is symmetric or skew-symmetric.

If A is a skew-symmetric and n ∈ N such that (Aⁿ)^T = λAⁿ, write the value of λ.

If A is a symmetric matrix and n ∈ N, write whether Aⁿ is symmetric or skew-symmetric or neither of these two.

If A is a skew-symmetric matrix and n is an even natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of these two.

If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of the two.

If A and B are symmetric matrices of the same order, write whether AB – BA is symmetric or skew-symmetric or neither of the two.

Write a square matrix which is both symmetric as well as skew-symmetric.

Find the value of x and y, if .

If , find x and y.

Find the value of x from the following: .

Find the value of y, if .

Find the value of x, if .

If matrix , write AA^T.

If , then find x.

If , find A + A^T.

If , then find a.

If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.

If is identity matrix, then write the value of α.

If , then write the value of k.

If I is the identity matrix and A is a square matrix such A² = A, then what is the value of (I + A)² – 3A?

If is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then find B.

If A is 2 × 3 matrix and B is a matrix such that A^TB and BA^T both are defined, then what is the order of B?

What is the total number of 2 × 2 matrices with each entry 0 or 1?

If , then find the value of y.

If a matrix has 5 elements, write all possible orders it can have.

For a 2 × 2 matrix A = [a_ij] whose elements are given by , write the value of a₁₂.

If , find the value of x.

If , then find matrix A.

If , find the value of b.

For what value of x, is the matrix a skew-symmetric matrix?

If matrix and A² = pA, then write the value of p.

If A is a square matrix such that A² = A, then write the value of 7A–(I + A)³, where I is the identity matrix.

If , find x – y.

If , find x.

If , write the value of a – 2b.

Write a 2 × 2 matrix which is both symmetric and skew-symmetric.

If , write the value of (x + y + z).

Construct a 2 × 2 matrix A = [a_ij] whose elements a_ij are given by .

If , then write the value of (x, y).

Matrix is given to be symmetric, find the values of a and b.

Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.

If , then write the order of matrix A.

If is written as A = P + Q, where as A = P + Q, where P is symmetric and Q is skew-symmetric matrix, then write the matrix P.

Let A and B be matrices of orders 3 × 2 and 2 × 4 respectively. Write the order of matrix AB.

If , n ∈ N, the A⁴ⁿ equals

If A and B are two matrices such that AB = A and BA = B, then B² is equal to

If AB = A and BA = B, where A and B are square matrices, then

If A and B are two matrices such that AB = B and BA = A, then A² + B² is equal to

If , then the least positive integral value of k is

If the matrix AB is zero, then

Let , then Aⁿ is equal to

If A, B and are square matrices or order 3, A is non-singular and AB = O, then B is a

If and , then AB is equal to

If , then Aⁿ (where n ∈ N) equals

If and and AB = I₃, then x + y equals

If and (A + B)² = A² + B², then values of a and b are

If is such that A² = I, then

If S = [s_ij] is a scalar matrix such that s_ij = k and A is a square matrix of the same order, then AS = SA = ?

If A is a square matrix such that A² = A, then (I + A)³ – 7A is equal to

If a matrix A is both symmetric and skew-symmetric, then

The matrix is

If A is a square matrix, then AA is a

If A and B are symmetric matrices, then ABA is

If and A = A^T, then

If A is 3 × 4 matrix and B is a matrix such that A^TB and BA^T are both defined. Then, B is of the type

If A = [a_ij] is a square matrix of even order such that a_ij = i² – j², then

If , then A^T + A = I₂, if

If is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is

Out of the following matrices, choose that matrix which is a scalar matrix:

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is

Which of the given values of x and y make the following pairs of matrices equal?

and,

If and , then the values of k, a, b, are respectively

If and , then B equals

The trace of the matrix is

If A = [a_ij] is scalar matrix of order n × n such that a_ij = k for all i, then trace of A is equal to

The matrix is a

The number of possible matrices of order 3 × 3 with each entry 2 or 0 is

If , then the value of x + y is

If A is a square matrix such that A² = I, then (A – I)³ + (A + I)³ – 7A is equal to

If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A – 2B is

If A is a matrix of order m × n and B is a matrix such that AB^T and B^TA are both defined, then the order of matrix B is

If A and B are matrices of the same order, then AB^T – B^TA is a

If matrix , where , then A² is equal to

If , then A – B is equal to

If A and B are square matrices of the same order, then (A + B)(A – B) is equal to

If and , then

The matrix is a

The matrix is