SOLUTION: 1.(a)given the matrix=A {2 3},what ar its dimension {0-1} {1 4} b)can these matrices be multiplied,why or why not? {2} {3 0} {4} {-2 3} {0} {1 2}

Dimensions are "down by across".  It is 3 elements down
and 2 elements across.  So it's a 3 by 2.

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