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.

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

Express the matrix A = 3 - 2 1 4 0 6 - 1 2 1 as the sum of a symmetric and a skew symmetric matrices.

For square matrix A , it can be expressed as the sum of symmetric and skew-symmetric matrix as follows A = 1 2 A + A ' + 1 2 A - A ' , where A ' is transpose matrix of A Given, A = 3 - 2 1 4 0 6 - 1 2 1 A ' = 3 4 - 1 - 2 0 2 1 6 1 Now, 1 2 A + A ' = 1 2 3 - 2 1 4 0 6 - 1 2 1 + 3 4 - 1 - 2 0 2 1 6 1 = 1 2 6 2 0 2 0 8 0 8 2 = 3 1 0 1 0 4 0 4 1 And 1 2 A - A ' = 1 2 3 - 2 1 4 0 6 - 1 2 1 - 3 4 - 1 - 2 0 2 1 6 1 = 1 2 0 - 6 2 6 0 4 - 2 - 4 0 = 0 - 3 1 3 0 2 - 1 - 2 0 ∴ A = 3 1 0 1 0 4 0 4 1 + 0 - 3 1 3 0 2 - 1 - 2 0 .

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 ]

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 = 2 3 - 1 4 as a sum of a symmetric and a skew-symmetric matrix.

Given matrix is A = 2 3 - 1 4 We need to express this matrix as a sum of a symmetric and a skew-symmetric matrix. We know that the transpose of matrix A is denoted by A ' , and is obtained by interchanging the rows and columns of the matrix A . Thus, A ' = 2 - 1 3 4 A + A ' = 2 3 - 1 4 + 2 - 1 3 4 = 4 2 2 8 1 2 A + A ' = 1 2 4 2 2 8 = 2 1 1 4 Let P = 2 1 1 4 We also observe that P = P ' ∴ P is a symmetric matrix. Now, A - A ' = 2 3 - 1 4 - 2 - 1 3 4 = 0 4 - 4 0 1 2 A - A ' = 1 2 0 4 - 4 0 = 0 2 - 2 0 Let Q = 0 2 - 2 0 Now, Q ' = 0 - 2 2 0 = - 0 2 - 2 0 = - Q ⇒ Q is a skew-symmetric matrix. Hence, A = 1 2 A + A ' + 1 2 A - A ' = P + Q , where P is a symmetric matrix, and Q is a skew-symmetric matrix. i.e., 2 3 - 1 4 = 2 1 1 4 + 0 2 - 2 0

Express 2 5 − 1 3 1 5 7 6 9 as sum of symmetric and skew-symmetric matrices.

Let A = 2 5 - 1 3 1 5 7 6 9 A T = 2 3 7 5 1 6 - 1 5 9 We know that A = 1 2 A + A T + 1 2 A - A T Now, 1 2 A + A T = 1 2 2 5 - 1 3 1 5 7 6 9 + 2 3 7 5 1 6 - 1 5 9 = 1 2 4 8 6 8 2 11 6 11 18 = 2 4 3 4 1 11 2 3 11 2 9 1 2 A - A T = 1 2 2 5 - 1 3 1 5 7 6 9 - 2 3 7 5 1 6 - 1 5 9 = 1 2 0 2 - 8 - 2 0 - 1 8 1 0 = 0 1 - 4 - 1 0 - 1 2 4 1 2 0 ⇒ 2 5 - 1 3 1 5 7 6 9 = 2 4 3 4 1 11 2 3 11 2 9 + 0 1 - 4 - 1 0 - 1 2 4 1 2 0 .