Solution of Chapter 3. Matrices (Mathematics Part-I Book)

In a matrix A =

(i) The order of the matrix,

(ii) The number of elements,

(iii) Write the elements a₁₃, a₂₁, a₃₃, a₂₄, a₂₃.

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

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

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

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

Construct a 3 × 4 matrix, whose elements are given by:

Construct a 3 × 4 matrix, whose elements are given by:

Find the values of x, y and z from the following equations:

Find the values of x, y and z from the following equations:

Find the values of x, y and z from the following equations:

Find the value of a, b, c and d from the equation:

A = [a_ij]_{m × n} is a square matrix, if

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

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

Le t

Find each of the following:

A + B

Let

Find each of the following:

A – B

Let

Find each of the following:

3A – C

Let

Find each of the following:

AB

Let

Find each of the following:

BA

Compute the following:

Compute the following:

Compute the following:

Compute the following:

Compute the indicated products.

Compute the indicated products.

Compute the indicated products.

Compute the indicated products.

Compute the indicated products.

Compute the indicated products.

If then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.

If then compute 3A – 5B.

Simplify

Find X and Y, if

Find X and Y, if

Find X, if

Find x and y, if

Solve the equation for x, y, z and t, if

If , find the values of x and y.

Given , find the values of x, y, z and w.

If show that F(x) F(y) = F(x + y).

Show that

Show that

Find A² – 5A + 6I, if

If, prove that A₃ – 6A² + 7A + 2I = 0

If , find k so that A² = kA – 2I

If and I is the identity matrix of order 2, show that

A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

Rs 1800

A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

Rs. 2000

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively.

The restriction on n, k and p so that PY + WY will be defined are:

If n = p, then the order of the matrix 7X – 5Z is:

Find the transpose of each of the following matrices:

Find the transpose of each of the following matrices:

Find the transpose of each of the following matrices:

If and, then verify that

(A + B)’ = A’ + B’,

If and, then verify that

(A – B)’ = A’– B’

If and, then verify that

(A + B). = A’ + B’

If and, then verify that

(A – B)’ = A’– B’

Ifand , then find (A + 2B)’

For the matrices A and B, verify that (AB)’ = B’A’, where

(i)

For the matrices A and B, verify that (AB)’ = B’A’, where

If , then verify that A’ A = I

If, then verify that A’ A = I

Show that the matrix is a symmetric matrix.

Show that the matrix is a skew symmetric matrix.

For the matrix , verify that

(A + A’) is a symmetric matrix

For the matrix , verify that

(A – A’) is a skew symmetric matrix

Find and , when

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

If A, B are symmetric matrices of same order, then AB – BA is a

If , and A + A’ = I, if the value of a is

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Using elementary transformations, find the inverse of each of the matrices.

Matrices A and B will be inverse of each other only if

Let, show that (aI + bA)ⁿ = aⁿI + na^n-1bA, where I is the identity matrix of order 2 and n ϵ N.

If , prove that .

If then prove that where n is any positive integer.

If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Find the values of x, y, z if the matrix satisfy the equation A’A =I.

For what values of

If show that A² – 5A + 7I =0.

Find x, if

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:

(a) If unit sale prices of x, y and z are Rs. 2.50, Rs. 1.50 and Rs. 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs. 2.00, Rs. 1.00 and 50 paise respectively. Find the gross profit.

Find the matrix X so that

If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABⁿ = BⁿA. Further, prove that (AB)ⁿ = AⁿBⁿ for all n ϵ N.

If is such that A � = I, then

If the matrix A is both symmetric and skew symmetric, then

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