I need help with Matrix and the directions in the book say
"Find the product. If it is not defined, state the reason.
The matrix is 

[7.3 1.5] 
[1.8   0] [-4.2 2.6 -8.7]
[2.9 3.2]

The dimensions of a matrix is always "down by across".
The first matrix is 3 down by 2 across, or 3×2. The second
matrix is 1 down by 3 across, or a 1×3 matrix.

Matrix multiplication is not commutative.

How to tell if you can multiply a pair of matrix on the
left of a multiplication sign by a matrix on the right of
a multiplication sign:

1. Determine the dimensions of the matrix on the left.
2. Determine the dimensions of the matrix on the right.
3. If the inner two numbers are the same, the multiplication
   can be made and the product matrix will have the dimensions
   of the two outer numbers.
4. If the inner two numbers are NOT the same, the multiplication
   cannot be made and there is no product matrix.

Examples:

A 3×4 matrix can be multiplied by a 4×2 matrix because the inner
dimensions are both the same, namely 4.  The product matrix will
be a 3×2 matrix, the two outer dimensionss.

A 4×3 matrix cannot be multiplied by a 2×4 matrix because the inner
dimensions, 3 and 2, are NOT both the same.  There is no product 
matrix.

A 5×2 matrix can be multiplied by a 2×5 matrix because the inner
dimensions are both the same, namely 2.  The product matrix will
be a 5×5 matrix, the two outer dimensions.

A 3×3 matrix cannot be multiplied by a 4×3 matrix because the inner
dimensions, 3 and 4, are NOT both the same.  There is no product 
matrix defined.

A 73×29 matrix can be multiplied by a 29×99 matrix because the inner
dimensions are both the same, namely 29.  The product matrix will
be a 73×99 matrix, the two outer dimensions.   

A 4×6 matrix cannot be multiplied by a 4×6 matrix because the inner
dimensions, 6 and 4, are NOT both the same.  There is no product
matrix defined.

Now for your answer:
You have a 3×2 matrix to be multiplied by a 1×3.  The inner dimensions
are 2 and 1. These are NOT the same so they cannot be multiplied.

If they had been reversed they could have been be multiplied:

If your problem had been given this way:

                [7.3 1.5] 
[-4.2 2.6 -8.7]·[1.8   0] 
                [2.9 3.2]

Then you could have multiplied them for you would have had a
1×3 multiplied by a 3×2.  The inner dimensions would have been
the same and the result would have been a 1×2 matrix, namely

[-51.21 -34.14]

But as it is, you have 

[7.3 1.5] 
[1.8   0] [-4.2 2.6 -8.7]
[2.9 3.2]

which is a 3×2 on the left and a 1×3 on the right,
and you cannot multiply them because the two inner 
dimensions are 2 and 1, which are not the same. So
the multiplication is not defined.

Edwin