# Precalculus : Multiplication of Matrices

## Example Questions

### Example Question #9 : Find The Product Of Two Matrices

We consider the matrices  and  below. We suppose that  and  are of the same size

What is the product ?

Explanation:

Note that every entry of the product matrix is the sum of  ( times) .

This gives  as every entry of the product of the two matrices.

### Example Question #10 : Find The Product Of Two Matrices

We will consider the two matrices

We suppose that  and  have the same size

What is ?

Explanation:

Note that when we multiply the first row by the first colum we get:  ( times), this gives the value of .

All other rows are zeros, and therefore we have zeros in the other entries.

### Example Question #51 : Matrices And Vectors

We consider the matrices  and  that we assume of the same size .

Find the product .

Explanation:

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.

### Example Question #52 : Matrices And Vectors

We consider the two matrices  and  defined below:

,

What is the matrix ?

We can't find the product

Explanation:

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:

### Example Question #53 : Matrices And Vectors

Let

and

What is the matrix ?

Product cannot be found.

Product cannot be found.

Explanation:

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.

### Example Question #11 : Find The Product Of Two Matrices

We consider the two matrices  and  given below, what is the simplest formula possible for  (assume that  and  have the same size).

We can't find the sum of the two matrices.

Explanation:

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:

### Example Question #61 : Matrices And Vectors

We consider the two matrice given below, find :

Explanation:

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:

### Example Question #62 : Matrices And Vectors

We recall the complex number  satisfies :

We define the matrix as follows:

Find the matrix .

We can't find this multiplication.

Explanation:

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:

### Example Question #63 : Matrices And Vectors

Multiply:

Explanation:

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:

Multiply