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- Yes. Let's consider the generic 2 by 2 matrix
 .
Let's look at the left side of the equation;
 .
Now let's look at the right side of the equation;
 .
Since a and each element of B are scalars, the order of
multiplication does not matter. Therefore, if
 ,
then AB = BA for any B for which matrix multiplication is
defined.
For the general case, A = aI, let's look at the left side of the equation.
The product AB = aIB = aB for any matrix B for which matrix multiplication
is defined. Therefore, a general element of this matrix is abij. Now,
let's look at the right side of the equation. We need to remember that a = aT because a is a scalar. The product
BA = BaI = Ba = (aTBT)T = (aBT)T. Each
element of the matrix aBT is abji, so the transpose of this matrix
has elements abij. Therefore, the two sides are equal.
- No. Unlike multiplication with real numbers, AB ≠ BA
in general.
There are occasions when AB = BA, but these occasions are very rare. In
fact, the only time that AB = BA for every B is when A is a scalar
multiple of the identity matrix. It is very important to remember that
AB is NOT, in general, equal to BA.
- If the dimensions are correct for multiplication,
A(BC) = (AB)C. We
call this the associative property of matrices. An example with correct
dimensions is matrix A is 4 by 3, matrix B is 3 by 2, and matrix C is
2 by 5. This product results in a 4 by 5 matrix. The associative property of
matrices becomes quite useful when you want to reduce the number of
multiplications performed. Refer to problem 5 for an example. The
number of multiplications performed becomes very important when you are
dealing with large matrices.
- Yes,
A(B + C) = AB + AC. This means that the distributive property holds
for matrices.
- Yes,
(AB)T = BTAT. If matrix A is 4 by 3 and matrix B is 3 by
5, AB is 4 by 5, so (AB)T is 5 by 4. Just by looking at dimensions, we
can tell that (AB)T ≠ ATBT
because the dimensions of ATBT tell
us that this multiplication cannot be performed. The dimensions of BTAT
are correct for matrix multiplication and give a resulting matrix that is
5 by 4. This is not a proof that
(AB)T = BTAT is true, but it is a good
indication. Work several examples to convince yourself that
(AB)T = BTAT.
- Yes,
A - B = -(B - A) if the dimensions of A and B are the same so
that subtraction is defined. This is true because
-(B - A) = -1(B - A) = -B + A = A - B.
- No. For example, if
and
then
but neither A nor B is a zero matrix. Remember that AB = 0 does NOT imply that either A = 0 or B = 0.
- Yes, ATA and AAT are always symmetric. Remember that
(AB)T = BTAT and that a matrix is symmetric if it is equal to its transpose. Let's look at the transpose of ATA; (ATA)T = AT(AT)T = ATA. Therefore, ATA is symmetric. The same procedure proves that AAT is symmetric;
(AAT)T = (AT)TAT = AAT.
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