MMULT

MMULT(MINVERSE(MMULT(TRANSPOSE(C5 E15),C5 E15)),MMULT(TRANS... 

MADD adds complex matrices 4 nd to give C MMULT multiplies complex matrices and to give C IDENT forms an N x Af identity matrix I CONJT takes the conjugate transpose of matrix ... 

In Figure 19-5, we enter the wavelengths in column A just to keep track of information. We will not use these wavelengths for computation. Enter the products eh for pure X in column B and eh for pure Y in column C. The array in cells B5 C6 is the matrix K. The Excel function MINVERSE(B5 C6) gives the inverse matrix, K-1. The function MMULT(matrix 1, matrix 2) gives the product of two matrices (or a matrix and a vector). The concentration vector, C, is equal to K 1 A, which we get with the single statement... 

To use the template in Figure 19-5, enter the coefficients, eb, measured from the pure compounds, in cells B5 C6. Enter the absorbance of the unknown in cells D5 D6. Highlight cells F5 F6 and type the formula =MMULT(MINVERSE(B5 C6), D5 D6j . Press CONTROL+SHIFT+ENTER on a PC or COMMAND(3 )+RETURN on a Mac. The concentrations [XI and [Y] in the mixture now appear in cells F5 F6. 

It is not necessary to give a matrix a name, so the expression =TRANSPOSE(Cll G17) is acceptable, and it is entirely possible to mix terminology, for example, =MMULT (X,B6 D9) will work, provided that die relevant dimensions are correct. 

It is possible to add and subtract matrices using + and —, but remember to ensure that the two (or more) mattices have the same dimensions as has the destination. It is also possible to mix matrix and scalar operations, so that the syntax =2 MMULT(X,Y) is acceptable. Furthermore, it is possible to add (or subtract) matrices consisting of a constant number, for example =Y +2 would add 2 to each element of Y. Other conventions for mixing matrix and scalar variables and operations can be determined by practice, although in most cases the result is what we would logically expect. 

Leave these cells selected then type =mmult(minverse(D2 J8),C2 C8) then simultaneously press Ctrl Shift Enter. 

The matrix is solved as described in Appendix H. The procedure is select cell B35, shift-arrow down to cell B41, type mmult(minverse(D22 J28),C22 C28) then simultaneously press Ctrl Shift Enter. This is the only time the matrix must be solved. 

Excel provides functions for the manipulation of arrays or matrices TRANSPOSE(array) returns the transpose of an array, MDETERM(array) returns the matrix determinant of an array, MINVERSE(array) returns the matrix inverse of an array, MMULT(arrayl, array2) returns the matrix product of two arrays and SUMPRODUCT(arrayI, array2,. ..) returns the sum of the products of corresponding array elements. These functions are discussed more fully in Chapter 9. 

Vector multiplication can be accomplished easily by the use of one of Excel s worksheet functions for matrix algebra, MMULT(inafr/x7, matrix2). For the matrices A and B defined above. 

If you use a worksheet function within VBA that returns an array, the lower array index will be 1. Such worksheet functions include LI NEST, TRANSPOSE, MINVERSE, MMULT. Other functions that return arrays include the VBA function Caller when used with a menu command or toolbutton. 

Related Functions MINVERSE, MMULT, SUMPRODUCT, TRANSPOSE... 

Up to 30 arrays may be included. All arrays must have the same dimensions. Related Functions MMULT, TRANSPOSE... 

Two of the functions under Math Trig are MINVERSE and MMULT. These functions have the obvious use. The inverse function, of course, must act on a square matrix, and creates a square matrix. To use it, first create the matrix, as illustrated in cells A1 C3 of Figure A.ll. 

Then in cell A6 type = MINVERSE(A1 C3). You can also insert the Al C3 by selecting the cells. Click on cell A6, press the shift key, and select the other comer of the matrix, C8. Press F2, the Ctrl-Shift-Enter. The inverse appears in A6 C8. The matrix multiplication is illustrated in Figure A. 11 by multiplying these two matrices together the result should be the identity matrix. Click on cell All, type =MMULT(A1 C3,A6 C8). Click on cell All, press the shift key, and select the other corner of the matrix, C13. Press F2, the Ctrl-Shift-Enter. The matrix multiplication appears in Al 1 C13. Indeed it is the identity matrix. 

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