# Precalculus : Matrices and Vectors

## Example Questions

### 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:

### Example Question #64 : Matrices And Vectors

Multiply

Explanation:

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

### Example Question #65 : Matrices And Vectors

Which of the following matrices can be multiplied?

Explanation:

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.

### Example Question #66 : Matrices And Vectors

Matrices  and  are shown above. Find the matrix product .

Explanation:

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,

### Example Question #65 : Matrices And Vectors

Calculate AB when

and .

Explanation:

In order to perform matrix multiplication, the number of columns in the first matrix has to be the same as the number of rows in the second column.

From here, we multiply each term in the first matrix's row by the first column in the second matrix. Continue in this fashion to get the product matrix.

### Example Question #66 : Matrices And Vectors

Evaluate:

Explanation:

Write the rule for multiplying a two by two matrix.  The result will be a two by two matrix.

Follow this rule for the given problem.

### Example Question #65 : Matrices And Vectors

Multiply the matrices:

Explanation:

In order to multiply these matrices we will need to consider the rows and columns for each matrix.

Both matrices have a dimension of .

The rule for multiplying matrices is where the number of columns of the first matrix must match the number of rows of the second matrix.

If the dimensions of the first matrix are , and the dimensions of the second matrix are , then we will get a dimension of  matrix as a result.  If the value of  are not matched, we cannot evaluate the product of the matrices.

### Example Question #68 : Matrices And Vectors

Find the product of A and B.