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Let us choose a generic matrix. We need to be careful when choosing a
generic matrix. Vectors and square matrices often have special properties,
so we will not use them unless they are specifically needed. Let us follow
our rules for transposes on this generic matrix.
Now let's form the transpose of AT using the same rules.
Notice that (AT)T = A. This is true for all matrices, but we have only proven it for 2 by 3 matrices. For a general proof, let us follow a general element of the matrix A, aij. Initially, it is in position (i,j) of matrix A. It is in position (j,i) of AT and in position (i,j) of matrix (AT)T.
This is true for every element of every matrix, so (AT)T = A is true in general.
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Yes. When a matrix is transposed, the columns become rows and the
rows become columns. If A = AT, the matrix must have the same number of rows as columns.
Remark 4 A counterexample
is an example that illustrates the statement in question is false.
When you want to prove that a statement is false, a counterexample
is sufficient. However, an example is not sufficient to prove that
a statement is true. For instance, you could use your father as an
example of the statement that "all humans are male" because he is
human and male. However, we know that there are humans that are not
male; our mother is a good countefexample to the statement "all humans
are male." Therefore, an example cannot be used to prove that a statement
is true. You would have to establish in some way that EVERY human
is male.
- No. A counterexample is
 . Since a12 ≠ a21, A is not symmetric.
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