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- Yes. Let's look at the two general matrices of dimensions 2 by 3
Looking at these general matrices should indicate to us that A + B = B + A
because the commutative law of addition for real numbers tells us that
aij + bij = bij + aij for any i and any j. Therefore, A + B = B + A
is true when the operations are defined (ie., when the matrices have the
same dimensions.) We proved this for 2 by 3 matrices and reasoned that it
would be true for matrices of other dimensions. We can prove that A + B = B + A
in general by looking at the general
(i,j)th element of each side of
the equation. The
(i,j)th element of A + B is
aij + bij and the
(i,j)th element of B + A is
bij + aij. Therefore, using the
commutative law of addition for real numbers, A + B = B + A.
- Yes,
(A + B)T = AT + BT. Let us look at a generic element from each
side of the equation. First, let's look at the left side of the equation. A
generic element of A + B would be
aij + bij. When this matrix is
transposed, the generic element is
aji + bji. Therefore, a generic
element of the left side of the equation is
aji + bji, which is
exactly the same as a generic element of the right side of the equation.
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