For any square matrix A with Real numbers. Prove that A + A1 is a symm

1.	For any square matrix A with Real numbers. Prove that A + A1 is a symmetric and A - A1 is a skew symmetric.
Answer» Let B = A + A¹ then B¹ = (A + A¹)¹ = A¹+ (A¹)¹ = A¹ + A= A + A¹ = B ∴ B¹ = B ∴B is symmetric matrix ∴ A + A¹ is a symmetric matrix. C = A – A¹ C¹ = (A – A¹)¹ = A¹ – (A¹)¹ = A¹ – A = -(A – A¹) = -C C¹ = -C is a skew-symmetric matrix. ∴ A – A¹ is a skew-symmetric matrix.

For any square matrix A with Real numbers. Prove that A + A1 is a symmetric and A - A1 is a skew symmetric.

Answer»

Let B = A + A¹ then

B¹ = (A + A¹)¹ = A¹+ (A¹)¹

= A¹ + A= A + A¹ = B

∴ B¹ = B ∴B is symmetric matrix

∴ A + A¹ is a symmetric matrix.

C = A – A¹

C¹ = (A – A¹)¹ = A¹ – (A¹)¹

= A¹ – A = -(A – A¹) = -C

C¹ = -C is a skew-symmetric matrix.

∴ A – A¹ is a skew-symmetric matrix.

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For any square matrix A with Real numbers. Prove that A + A1 is a symmetric and A - A1 is a skew symmetric.

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