Question

Linear Algebra: Determine whether the matrices At + A and At - A are symmetric or skew-symmetric.

Answer 1

\[A^{T} - A, A^{T} + A\]
"T" referring to the transpose of A

Answer 2

what do u get when u do the following?
\[ \large (A^t-A)^t= \]
and
\[ \large (A^t+A)^t= \]

Answer 3

remember this
\[ \large (A+B)^t=A^t+B^t \]
\[ \large (rA)^t=r\cdot A^t \]

Answer 4

\[(A^{T} - A)^{T} = A - A^{T}, (A^{T} + A)^{T} = A + A^{T}, \]

Answer 5

great. now u know this definitions:
\[ \large A\text{ is symmetric if }A^t=A \]
\[ \large A\text{ is skew-symmetric if }A^t=-A \]

Answer 6

then I just add or subtract to both sides?

Answer 7

no

Answer 8

u got this
\[ \large (A^t+A)^t=(A^t)^t+A^t=A+A^t=A^t+A \]
so according with the definition i gave u. is this matrix symmetric or skew-symmetric?

Answer 9

well @jcd2012 ?

Answer 10

im trying to formulate an answer, you're basically saying I should think as the transpose of the sum and differences as one matrix being transposed, so it meets the definition?

Answer 11

exactly

Answer 12

then by the definition, (A^T - A) is skew-symmetric and (A^T + A) is symmetric

Answer 13

right?

Answer 14

right.