If the matrix A = 0 a - 3 2 0 - 1 b 1 0 is skew symmetric, find the values of a and b .

Given that, A = 0 a - 3 2 0 - 1 b 1 0 And the matrix A is given to be skew-symmetric matrix ∴ A T = - A ∴ 0 2 b a 0 1 - 3 - 1 0 = - 0 a - 3 2 0 - 1 b 1 0 ⇒ 0 2 b a 0 1 - 3 - 1 0 = 0 - a 3 - 2 0 1 - b - 1 0 Two matrices A and B are said to be equal if they are of same order and their corresponding elements are equal. Therefore, - a = 2 and b = 3 ⇒ a = - 2 and b = 3 . Hence, the required values are: a = - 2 and b = 3

If A = 5 a b 0 and A is symmetric matrix, show that a = b .

Given that A = 5 a b 0 and A is symmetric matrix. A square matrix that is equal to its transpose is called a symmetric matrix. For example, a square matrix A = a i j is symmetric if and only if a i j = a j i for all values of i and j Now, transpose of A is A ' = 5 b a 0 According to the definition of symmetric matrix, A = A ' ⇒ 5 a b 0 = 5 b a 0 On equating we have a = b Hence, proved.

Show that A + A ' is symmetric matrix, if A = 2 4 3 5

Given, A = 2 4 3 5 We need to find: A + A ' We know that A ' is the notation for Transpose of A . And, transpose of a matrix is obtained by interchanging the rows and columns of the original matrix. Thus, A ' = 2 3 4 5 Now, A + A ' = 2 4 3 5 + 2 3 4 5 = 2 + 2 4 + 3 3 + 4 5 + 5 = 4 7 7 10 . . . . i Now, let us find : A + A ' ' A + A ' ' = 4 7 7 10 . . . . . i i Since A + A ' ' = A + A ' Hence, A + A ' is a symmetric matrix.

Express the matrix A = - 1 5 1 2 3 4 7 0 9 as the sum of a symmetric and a skew-symmetric matrix.

Given matrix is A = - 1 5 1 2 3 4 7 0 9 We know that the transpose of a matrix is denoted by A ' , and is obtained by interchanging rows and columns of matrix A . Thus, A ' = - 1 2 7 5 3 0 1 4 9 Now, A + A ' = - 1 5 1 2 3 4 7 0 9 + - 1 2 7 5 3 0 1 4 9 = - 1 - 1 5 + 2 1 + 7 2 + 5 3 + 3 4 + 0 7 + 1 0 + 4 9 + 9 = - 2 7 8 7 6 4 8 4 18 Now, 1 2 A + A ' = 1 2 - 2 7 8 7 6 4 8 4 18 = - 1 7 2 4 7 2 3 2 4 2 9 Let P = 1 2 A + A ' ⇒ P = - 1 7 2 4 7 2 3 2 4 2 9 . . . i Now, we can observe that P ' = P Thus, the matrix P is symmetric. Now, let us find: A - A ' A - A ' = - 1 5 1 2 3 4 7 0 9 - - 1 2 7 5 3 0 1 4 9 = - 1 + 1 5 - 2 1 - 7 2 - 5 3 - 3 4 - 0 7 - 1 0 - 4 9 - 9 = 0 3 - 6 - 3 0 4 6 - 4 0 Now, 1 2 A - A ' = 1 2 0 3 - 6 - 3 0 4 6 - 4 0 = 0 3 2 - 3 - 3 2 0 2 3 - 2 0 Let Q = 0 3 2 - 3 - 3 2 0 2 3 - 2 0 . . . . . i i Also, we can observe that Q ' = - Q Now, P + Q = 1 2 A + A ' + 1 2 A - A ' = A Hence, we can express the given matrix A as the sum of a symmetric and a skew-symmetric matrix as follows: - 1 5 1 2 3 4 7 0 9 = - 1 7 2 4 7 2 3 2 4 2 9 + 0 3 2 - 3 - 3 2 0 2 3 - 2 0 [From i , and i i ]