1. (a) Given the matrix ,
what are its dimensions?
Dimensions are "down by across". It is 3 elements down
and 2 elements across. So it's a 3 by 2.
(b) Can these matrices be multiplied? Why or why not?
No they can't, because the matrix on the left is a 3 by 1 matrix and the
matrix on the right is a 3 by 2 matrix.
You can only multiply a "p by q" matrix on the left by a "q by r" matrix
on the right and get a "p by r" matrix.
(p by q)×(q by r) = (p by r)
The "inner" dimensions q must be the same, and if so, then the product
matrix will have the "outer" dimensions, p by r. Examples:
A 6 by 5 matrix on the left can be multiplied by a 5 by 9 matrix on the right
to give a 6 by 9 matrix, because the inner dimensions are both 5 and the outer
dimensions are 6 by 9.
(6 by 5)×(5 by 9) = (6 by 9)
A 3 by 1 matrix on the left can be multiplied by a 1 by 3 matrix on the right
to give a 3 by 3 matrix, because the inner dimensions are both 1 and the outer
dimensions are 3 by 3.
(3 by 1)×(1 by 3) = (3 by 3)
A 1 by 3 matrix on the left can be multiplied by a 3 by 1 matrix on the right
to give a 1 by 1 matrix, because the inner dimensions are both 3 and the outer
dimensions are 1 by 1.
(1 by 3)×(3 by 1) = (1 by 1)
A 58 by 29 matrix on the left can be multiplied by a 29 by 73 matrix on the
right to give a 58 by 73 matrix, because the inner dimensions are both 29 and
the outer dimensions are 58 by 73.
(58 by 29)×(29 by 73) = (58 by 73)
However, in your problem (b), a 3 by 1 matrix on the left cannot be multiplied
by a 3 by 2 matrix on the right, because the inner dimensions are 1 and 3 are
they are not the same.
(3 by 1)×(3 by 2)
is not defined because the inner dimensions are not the same.
If you had been given these matrices instead:
then you would have had a 1 by 3 matrix on the left to multiply by a 3 by 2
matrix on the right, and they could be multiplied because the inner dimensions
are both 3. The product matrix would be a 1 by 2 matrix because they would be
the outer dimensions.
(1 by 3)×(3 by 2) = (1 by 2)
However if you wrote the second one on the left and the first one on the right
like this:
you could not multiply them because the inner dimensions would not be the same:
(3 by 2)×(1 by 3 is not defined.
The commutative principle does not hold for matrix multiplication like it
holds for ordinary number multiplication.
That is when you multiply 2×3 you get 6, which is the same as when you
multiply 3×2. So the commutative principle holds for ordinary number
multiplication.
Notice this however. When you multiply
,
and they can be multiplied, because
(2 by 2)×(2 by 2) = (2 by 2)
the inner dimensions are the same,
you get this 2 by 2 matrix
But when you multiply
,
and they can be multiplied, too, because
(2 by 2)×(2 by 2) = (2 by 2)
the inner dimensions are the same,
you get this 2 by 2 matrix
So if you reverse the order of two matrices, even if they can still be
multiplied, you don't get the same answer.
Edwin