Posts Tagged ‘Matrix’

Problem 1179

Is the following transition matrix regular?

A = \begin{bmatrix}  1 & 0  \\  0.75& 0.25  \end{bmatrix}

 

Solution:-

 

A transition matrix is regular if same power of the matrix contains all positive entries.

The given matrix is A, or A1. Since 0 is not a positive number, A1 does not contain all positive entries.

Thus, other powers of a must be examined to determine whether or not A is regular.

Next, look at A2, which if found by multiplying A by itself using matrix multiplication.

A2 = \begin{bmatrix}  1 & 0  \\  0.9375& 0.0625  \end{bmatrix}

 

Notice that both A and A2 contain only one zero and in both matrices this zero is in row 1 and column 2. For any transition matrix P, if all zeros occur in the identical places in both Pn and Pn+1 for any n, they will appear in those places for all higher powers of P. Therefore there would be no power of P that contain all positive entries, so P is  not regular.

Thus, the given transition matrix A is nor regular because the only 0 occurs in the identical place in both A1 and A2, which means there is no power of A that contains all positive entries.

Problem 1064

Find the size of the matrix. Determine if it is a square, column, or row matrix.

\begin{bmatrix}  -2 & 3\\  0 & -2\\  3& -4  \end{bmatrix}

 

Solution:-

 

To find the size of the matrix, count the number of row and the number of columns.

\begin{bmatrix}  -2 & 3\\  0 & -2\\  3& -4  \end{bmatrix}

The matrix has 3 rows and 2 columns.

The number of rows is given first, and then the number if columns.

It is a 3 × 2 matrix.

Next, determine if the matrix is a column, row, or square matrix. It will be a column matrix if it consists of just one column, a row matrix if it consists of just one row, or a square matrix if the number of rows and columns are equal.

The matrix is of no special type.

 

 

Matrix Row Operations

Matrix Row Operations

 

a. Any two rows of the matrix may be interchanged.

 

b.  The numbers in any row may be multiplied by any nonzero real

 

c.  Any row may be transformed by adding to the numbers of the row the product of a real number and the corresponding numbers of another row.

Examples of these row operations follow.

 

Row operation 1:

\begin{bmatrix}  2 &3  &9 \\  4&8  &-3 \\  1& 0 & 7  \end{bmatrix} becomes \begin{bmatrix}  1 & 0 &7 \\  4& 8 & -3\\  2& 3 & 9  \end{bmatrix} . Interchange row 1 and row 3.

 

Row operation 2:

 

\begin{bmatrix}  2 &  3&9 \\  4 & 8 &-3 \\  1& 0 & 7  \end{bmatrix} becomes \begin{bmatrix}  6 &  9&27 \\  4& 8 &-3 \\  1&0  & 7  \end{bmatrix}. Multiply the number in row 1 by 3.

 

Row operation 3:

 

\begin{bmatrix}  2 & 3 &9 \\  4 & 8 & -3\\  1& 0 & 7  \end{bmatrix} becomes \begin{bmatrix}  0 & 3 &-5 \\  4& 8 & -3\\  1& 0 & 7  \end{bmatrix}. Multiply the number in row 3 by -2; add then to the corresponding numbers in row 1.

The third row operation corresponds to the way we eliminated a variable from a pair of equations in the previous sections.