Scalar quantity matrix algebra

There is no matrix version of simple division, as with scalar quantities. Rather, the inverse of a matrix (A-1), which exists only for square matrices, is the closest analog to a divisor. An inverse matrix is defined such that AA"1 = A-1 A = I (all three matrices are n X n). In scalar algebra, the equation a-b = c can be solved for b by simply multiplying both sides of the equation by la. For a matrix equation, the analog of solving... 

The computation of the fluxes from either of Eqs. 8.3.24 necessarily involves an iterative procedure (except for the special cases discussed above), partly because the themselves are needed for the evaluation of the matrix of correction factors and also because an explicit relation for the matrix [0] cannot be derived as a generalization of Eq. 8.2.16 for binary mass transfer there is no requirement in matrix algebra for the matrices [FFq] be equal to each other even though the fluxes calculated from both parts of these equations must be equal. Indeed, these two matrices will be equal only in the case of vanishingly small mole fraction differences (yg Tg) and vanishingly small mass transfer rates. In almost all cases of interest these two matrices are quite different. An explicit solution was possible for binary systems only because all matrices reduce to scalar quantities. 

As in the case of scalar quantities, matrices can also be added, subtracted, multiplied and so forth, but special rules of matrix algebra must be followed. 

The quantities in the two-dimensional list are called matrix elements. Each matrix element has two subscripts, one for the row and one for the column. The brackets written on the left and right are part of the notation. If a matrix has the same number of rows as columns m = n), it is a square matrix. A vector in ordinary space can be represented as a list of three Cartesian components, which is a matrix with one row and three columns. We call this a row vector. A vector can also be represented by a column vector with three rows and one column. We can also define row vectors and column vectors with more than three elements when they apply to something other than ordinary space. Just as there are types of algebra for scalars, vectors, and operators, there is a well-defined matrix algebra. 

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