Class 12 RD Sharma Solutions- Chapter 5 Algebra of Matrices - Exercise 5.5

Exercise 5.5 in Chapter 5 of RD Sharma's Class 12 Mathematics focuses on the concept of rank of a matrix. This is a fundamental concept in linear algebra that helps in understanding the structure and properties of matrices, as well as their applications in solving systems of linear equations.

Question 1: 

Solution:

Given:

 

Consider,

From equation (1) and (2) it can be seen that,

A skew-symmetric matrix is a square matrix whose transpose equal to its negative, that is,

X = −X^T

So, A − A^T is a skew-symmetric.

Question 2: 

Solution:

Given:

Consider,

From equation (1) and (2) it can be seen,

A skew-symmetric matrix is a square matrix whose transpose equals its negative, that is,

X = −X^T

Thus, A − A^T is a skew-symmetric matrix.

Question 3: 

Solution:

Given:

As we know that A = [a_ij]_m×n is a symmetric matrix if a_ij = a_ji

Thus,

x = a₁₃ = a₃₁ = 4

y = a₂₁ = a₁₂ = 2

z = a₂₂ = a₂₂ = z

t = a₃₂ = a₂₃ = −3

Hence, x = 4, y = 2, t = −3 and z can have any value.

Question 4: 

Solution:

Given: