# If n skew-symmetric matrices of same order are ${A_1},{A_2},.......................{A_{2n - 1}}$, then $B = \sum\limits_{r = 1}^n {\left( {2r - 1} \right){{\left( {{A_{2r - 1}}} \right)}^{2r - 1}}} $will be

$

(a){\text{ symmetric}} \\

(b){\text{ skew - symmetric}} \\

(c){\text{ neither symmetric now - symmetric}} \\

(d){\text{ data not adequate }} \\

$

Answer

Verified

379.2k+ views

Hint – In this question it is given as ${A_1},{A_2},.......................{A_{2n - 1}}$ are n skew-symmetric matrices. A skew-symmetric matrix is one whose transpose is equal to a matrix multiplied with a negative sign that is${B^T} = - B$, use this condition while evaluating the submission to check whether it satisfies the options given in the question or not.

Complete step-by-step answer:

It is given that ${A_1},{A_2},.......................{A_{2n - 1}}$ are n skew-symmetric matrices of the same order.

So, we have to find out $B = \sum\limits_{r = 1}^n {\left( {2r - 1} \right){{\left( {{A_{2r - 1}}} \right)}^{2r - 1}}} $will be.

Now as we know the condition of skew-symmetric matrices of same order is

$ \Rightarrow {A_1}^T = - {A_1},{A_3}^T = - {A_3},.........................{A_{2n - 1}}^T = - {A_{2n - 1}}$ ………………….. (1)

[Where T is the transpose of the matrix]

Now expand the summation (from r = 1 to n) we have,

$ \Rightarrow B = \sum\limits_{r = 1}^n {\left( {2r - 1} \right){{\left( {{A_{2r - 1}}} \right)}^{2r - 1}}} $

$ \Rightarrow B = {A_1} + 3{\left( {{A_3}} \right)^3} + 5{\left( {{A_5}} \right)^5} + ................. + \left( {2n - 1} \right){\left( {{A_{2n - 1}}} \right)^{2n - 1}}$………………. (2)

Now take transpose of matrix B we have,

$ \Rightarrow {B^T} = {A_1}^T + 3{\left( {{A_3}^T} \right)^3} + 5{\left( {{A_5}^T} \right)^5} + ................. + \left( {2n - 1} \right){\left( {{A_{2n - 1}}^T} \right)^{2n - 1}}$

Now from equation (1) we have,

$ \Rightarrow {B^T} = - {A_1} + 3{\left( { - {A_3}} \right)^3} + 5{\left( { - {A_5}} \right)^5} + ................. + \left( {2n - 1} \right){\left( { - {A_{2n - 1}}} \right)^{2n - 1}}$

Now take (-) common we have,

$ \Rightarrow {B^T} = - \left[ {{A_1} + 3{{\left( {{A_3}} \right)}^3} + 5{{\left( {{A_5}} \right)}^5} + ................. + \left( {2n - 1} \right){{\left( {{A_{2n - 1}}} \right)}^{2n - 1}}} \right]$

Now from equation (2) we have,

$ \Rightarrow {B^T} = - B$

Which is the condition of skew-symmetric.

So, the matrix B is a skew-symmetric matrix.

Hence option (b) is correct.

Note – Whenever we face such types of problems the key concept is to use the gist of the basic definition of symmetric and skew-symmetric matrix. A symmetric matrix is one which even after transposed gives us the same matrix. Use these concepts of symmetric and skew-symmetric matrix to get the right option for the question.

Complete step-by-step answer:

It is given that ${A_1},{A_2},.......................{A_{2n - 1}}$ are n skew-symmetric matrices of the same order.

So, we have to find out $B = \sum\limits_{r = 1}^n {\left( {2r - 1} \right){{\left( {{A_{2r - 1}}} \right)}^{2r - 1}}} $will be.

Now as we know the condition of skew-symmetric matrices of same order is

$ \Rightarrow {A_1}^T = - {A_1},{A_3}^T = - {A_3},.........................{A_{2n - 1}}^T = - {A_{2n - 1}}$ ………………….. (1)

[Where T is the transpose of the matrix]

Now expand the summation (from r = 1 to n) we have,

$ \Rightarrow B = \sum\limits_{r = 1}^n {\left( {2r - 1} \right){{\left( {{A_{2r - 1}}} \right)}^{2r - 1}}} $

$ \Rightarrow B = {A_1} + 3{\left( {{A_3}} \right)^3} + 5{\left( {{A_5}} \right)^5} + ................. + \left( {2n - 1} \right){\left( {{A_{2n - 1}}} \right)^{2n - 1}}$………………. (2)

Now take transpose of matrix B we have,

$ \Rightarrow {B^T} = {A_1}^T + 3{\left( {{A_3}^T} \right)^3} + 5{\left( {{A_5}^T} \right)^5} + ................. + \left( {2n - 1} \right){\left( {{A_{2n - 1}}^T} \right)^{2n - 1}}$

Now from equation (1) we have,

$ \Rightarrow {B^T} = - {A_1} + 3{\left( { - {A_3}} \right)^3} + 5{\left( { - {A_5}} \right)^5} + ................. + \left( {2n - 1} \right){\left( { - {A_{2n - 1}}} \right)^{2n - 1}}$

Now take (-) common we have,

$ \Rightarrow {B^T} = - \left[ {{A_1} + 3{{\left( {{A_3}} \right)}^3} + 5{{\left( {{A_5}} \right)}^5} + ................. + \left( {2n - 1} \right){{\left( {{A_{2n - 1}}} \right)}^{2n - 1}}} \right]$

Now from equation (2) we have,

$ \Rightarrow {B^T} = - B$

Which is the condition of skew-symmetric.

So, the matrix B is a skew-symmetric matrix.

Hence option (b) is correct.

Note – Whenever we face such types of problems the key concept is to use the gist of the basic definition of symmetric and skew-symmetric matrix. A symmetric matrix is one which even after transposed gives us the same matrix. Use these concepts of symmetric and skew-symmetric matrix to get the right option for the question.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What are the measures one has to take to prevent contracting class 12 biology CBSE

Suggest some methods to assist infertile couples to class 12 biology CBSE

Amniocentesis for sex determination is banned in our class 12 biology CBSE

Trending doubts

State Gay Lusaaccs law of gaseous volume class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

What is pollution? How many types of pollution? Define it

Change the following sentences into negative and interrogative class 10 english CBSE

Which is the tallest animal on the earth A Giraffes class 9 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE