In using Markov analysis to predict future market shares or future states, matrices are multiplied. To multiply matrices in Excel, we use MMULT as follows:

1. Highlight all the cells that will contain the resulting matrix.

2. Type $=\text{MMULT}\left(matrix1, matrix2\right),$ where matrix 1 and matrix 2 are the cell ranges for the two matrices being multiplied.

3. Instead of just pressing Enter, hold down the Ctrl and Shift keys and then press Enter.

Pressing $\text{Ctrl}+\text{Shift}+\text{Enter}$ is used to indicate that a matrix operation is being performed so that all cells in the matrix are changed accordingly.

Program 14.3A shows the formulas for the grocery store example from Section 14.2. We have entered the time period in column A for reference only, as we will be computing the state probabilities through time period 6. The state probabilities (cells B6 through D6) and the matrix of transition probabilities (cells E6 through G8) are entered as shown. We then use matrix multiplication to find the state probabilities for the next period. Highlight cells B7, C7, and D7 (this is where the resulting matrix will be) and type $=\text{MMULT}\left(\text{B}6\text{:D}6\text{,E}6\text{:G}8\right),$ as shown in the table. Then press $\text{Ctrl}+\text{Shift}+\text{Enter}$ (all at one time), and this formula is put in each of the three cells that were highlighted. When you have done this, Excel places { } around this formula in the box at the top of the screen (not shown). We then copy cells B7, C7, and D7 to rows 8 through 12, as shown. Program 14.3B shows the result and the state probabilities for the next six time periods.

Excel can be used to find the fundamental matrix that is used to predict future conditions when absorbing states exist. The MINVERSE function is used in the accounts receivable example from Section 14.6 of this chapter. Program 14.4A shows the formulas, and Program 14.4B provides the results. Remember that the entire range of the matrix (D12 through E13) is highlighted before the MINVERSE function is entered. Also remember to press $\text{Ctrl}+\text{Shift}+\text{Enter}$ all at once.

Matrix addition and subtraction can be handled in a fashion similar to the methods just described. In Program 14.4A, the I – B matrix was computed using a matrix method. First, we highlighted cells D9 through E10 (where the result is to be). Then we typed the formula seen in these cells and pressed $\text{Ctrl}+\text{Shift}+\text{Enter}$ (all at once), causing this formula to be entered in each of these cells. Excel then computes the appropriate values, as shown in Program 14.4B.