An easy way to find this is to note that ; therefore, we can find  by squaring  and squaring the result. 

Matrix multiplication is worked row by column - each row in the former matrix is multiplied by each column in the latter by adding the products of elements in corresponding positions, as follows:

Now square this:

 is a diagonal matrix, so it can be raised to a power by raising the individual entries to that power:

By DeMoivre's Theorem, 

.

We will need to rewrite the entries in the matrix as sums, not differences. We can do this by noting that the cosine and sine functions are even and odd, respectively, so  can be rewritten as

Applying DeMoivre's Theorem,

The coterminal angle for both  and  is , so

, a column matrix with ten entries, is a  matrix; , its transpose, is a  matrix. 

If  and  are  and  matrices, respectively, then the product  is a  matrix. Therefore,  is a  matrix - a matrix with a single entry.

, a column matrix with ten entries, is a  matrix; , its transpose, is a  matrix. 

If  and  are  and  matrices, respectively, then the product  is a  matrix. Therefore,  is a  matrix.

 and  are both elementary matrices, in that each can be formed from the (four-by-four) identity matrix   by a single row operation.  

Since  differs from  in that the entry  is in Row 2, Column 2, the row operation is . Since  differs from  in that entry  is in Row 3, Column 2, the row operation is .

Premultiplying a matrix by an elementary matrix has the effect of performing that row operation on the matrix. Looking at  as :

Premultiply  by  by performing the operation :

Premultiply  by  by performing the operation :

This is the correct product.

All of the matrices are diagonal, so the seventh power of each can be determined by simply taking the seventh power of the individual entries in the main diagonal. Also, note that each entry in each choice is of the form 

.

By DeMoivre's Theorem, for any real ,

Combining these ideas, we can take the seventh power of each matrix and determine which exponentiation yields the identity.

If 

,

then 

If 

,

then 

If 

,

then 

If 

,

then 

 is the only possible matrix value of  among the choices.

 and  are both elementary matrices, in that each can be formed from the (four-by-four) identity matrix   by a single row operation. It also holds that each can be formed from  by a single column operation.

Since  differs from  in that the entry  is in Row 2, Column 2, the column operation is . Since  differs from  in that entry  is in Row 3, Column 2, the column operation is .

Postmultiplying a matrix by an elementary matrix has the effect of performing that column operation on the matrix. Looking at  as :

 Postmultiply  by  by performing the operation :

Postmultiply  by  by performing the operation :

.

This is the correct product.

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (2x2). Next, we notice that matrix A is the identity matrix. Any matrix multiplied by the identity matrix remains unchanged.  

Each column of a stochastic matrix has entries whose sum is 1. For each column in , set the sum of the entries equal to 1; it is easiest to work with Columns 1, 2, and 3 in that order, for reasons that become apparent:

Column 1: 

Column 2: 

It is known that , so substitute:

Column 3: 

It is known that , so substitute:

This is not as daunting a task as it seems. 

First, find :

Multiply rows of  by columns of  by adding the products of corresponding elements, as follows:

Find  similarly:

This suggests a pattern:

 for any .

This can be confirmed by mathematical induction as follows:

Suppose .

Then 

Setting :

.