Note that multiplying every row of  by the first column of  gives .

Mutiplying every row of  by the second column of  gives .

Now the remaining columns are columns of zeros, and therefore this product gives zero in every row-column product.

Knowing these three aspects we get the resulting matrix.

The first matrix is (4x1) and the second matrix is (1x3). We can perform the matrix multiplication in this case. The resulting matrix is (4x3).

The first entry in the formed matrix is on the first row and the first column.

It is coming from the product of the first row of A and the first column of B.

This gives .We continue in this fashion.

The entry (4,3) is coming from the 4th row of A and the 3rd column of B.

This gives . To obtain the whole matrix we need to remember that any entry on AB say(i,j) is coming from the product of the rom i from A and the column j of B.

After doing all these calculations we obtain:

We note first that A is 4x4 , B is 4x1. 

To be able to do BA the number of columns of B must equal the number of rows

of A.

Since the number of columns of B is 1 and the number of rows of A is 4, we do not have equality and therefore we can't have the product BA.

Since we are assuming that the two matrices have the same size, we can perform the matrices addition.

We know that when adding matrices, we add them componenwise. Let (i,j) be any entry of the addition matrix. We add the entry from A to the entry from B:

Since the entries from A are the same and given by ln(2) and the entries from B are the same and given by ln(3), we add these two to obtain :

ln(2)+ln(3) and by the properties of the logarithm we have ln(2)+ln(3)=ln(2x3)=ln(6).

Therefore our matrix is given by:

The number of columns of A is equal to the number of rows of B. Therefore we can perform this operation.

Any entry of the matrix product is the result of the sum of the  product of the elements of the row of A with the colum of of B. To obtain the first entry of the matrix product, we use the the first row of A and the first column of B, multiplying componentwise and adding. Doing this operation for each entry,

we obtain our matrix:

We can treat i as a scalar. To do this multiplication all we need to do is to multply each entry of the matrix by i.

We see that when we multiply each entry of the matrix by i, we obtain  and we know that .

This means that all the entries are equal to same scalar . Now placing all these scalar in the matrix entries we obtain:

To find the product of 2 matrices, first line up the first row of the left matrix with the first column of the right matrix. Multiply the first, second, and third entries and then add them together.

Next, line up the second row of the left matrix with the second column of the right matrix. Then the first row and the second column, and finally the second row and the first column:

To find the product, line up the rows of the left matrix individually with the one column in the right matrix:

The size of every matrix can be written in the form rows x cols. The following matrix is of the size 2 x 1 because it has 2 rows and 1 column.

For two matrices to be able to be multiplied, their sizes must line up that the number of columns in the first matrix is equal to the number of rows of the first matrix. For example:

So these two matrices can be multiplied. However, if the case were such that:

Here, the # of columns in the first matrix does not line up with the # of rows in the second matrix, so the two matrices cannot be multiplied.

First, note that the order of the matrix multiplication is important . Multiplication of two matrices is possible only if the number of columns of the first matrix  is equal to the number of rows of the second matrix . Both  and  are  matrices (2 rows and 2 columns, respectively). Thus,  is possible  since the number of columns of  (2) equals to the number of rows of  (2). Furthermore, the size of  is equal to the number of rows of  and the number of columns of  . 

To avoid confusion, I will use the notation , , and  to denote the constituents of matrices , , and , respectively. For example,  refers to the constituent in  that is in row 1 and column 2. The general version of the three matrices are shown below:

Using the rules of multiplying two matrices, the definition of  is shown below:

Thus,