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Question: If A is a squared matrix, then prove that \(A + A'\) is symmetric and \(A - A'\) is skew-symmetric....

If A is a squared matrix, then prove that A+A′A + A' is symmetric and A−A′A - A' is skew-symmetric.

Explanation

Solution

Hint – In this question use the concept that if A matrix is symmetric then A′=AA' = A and if A matrix is skew-symmetric then A′=−AA' = - A, where A′A' is the transpose of A matrix. Apply this same concept on A+A′A + A' matrix and A−A′A - A' matrix by taking transpose for them. This will help get the right answer.

Complete step-by-step answer:
As we know that if A is n squared matrix then the matrix is symmetric if and only if the transpose of a matrix (A) is the same as the given matrix.
⇒A′=A\Rightarrow A' = A, where A′A' is the transpose of A matrix.
And we all know that if A is a squared matrix then the matrix is skew symmetric if and only if the transpose of a matrix (A) is negative times the given matrix.
⇒A′=−A\Rightarrow A' = - A
Now we have to prove that A+A′A + A' is symmetric and A−A′A - A' is skew symmetric.
⇒(i)(A+A′)′=A+A′\Rightarrow \left( i \right){\left( {A + A'} \right)^\prime } = A + A', (ii)(A−A′)′=−(A−A′)\left( {ii} \right){\left( {A - A'} \right)^\prime } = - \left( {A - A'} \right)
Now take L.H.S of first equation
⇒(A+A′)′\Rightarrow {\left( {A + A'} \right)^\prime }
Now apply the transpose according to property (A+B)′=A′+B′{\left( {A + B} \right)^\prime } = A' + B' so we have,
⇒(A+A′)′=A′+(A′)′=A′+A\Rightarrow {\left( {A + A'} \right)^\prime } = A' + {\left( {A'} \right)^\prime } = A' + A, [∵(A′)′=A]\left[ {\because {{\left( {A'} \right)}^\prime } = A} \right]
= R.H.S
Now take L.H.S of second equation we have,
⇒(A−A′)′\Rightarrow {\left( {A - A'} \right)^\prime }
Now apply transpose according to property (A+B)′=A′+B′{\left( {A + B} \right)^\prime } = A' + B' so we have,
⇒(A−A′)′=A′−(A′)′=A′−A=−(A−A′)\Rightarrow {\left( {A - A'} \right)^\prime } = A' - {\left( {A'} \right)^\prime } = A' - A = - \left( {A - A'} \right), [∵(A′)′=A]\left[ {\because {{\left( {A'} \right)}^\prime } = A} \right]
= R.H.S
Hence proved.

Note – Transpose of a matrix is obtained by flipping of rows and columns that is first row is interchanged with first column, similarly second row with second column and the nth row of the matrix with nth column. If the row and column interchange result in formation of the same matrix than it is said to be a symmetric matrix and if the row and column interchange of the matrix result in formation of a matrix which is multiplied with the negative sign in original one than the matrix is said to be a skew-symmetric matrix.