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11) AikB kj = 1,2, .. 0' m;j = 1,2,. 0" n). Note the compatibility requirement, namely, that A must have the same number of columns as B has rows. 14) for all matrices over f/i of appropriately compatible sizes. 15) 22 2. 13) is thereby proved. There is a matrix generalization of the multiplicative identity 1 in :F. 16) IA = A = AI has a unique n x n matrix solution I over:F holding for all n x n A over :F; namely, (I)ij =. bijwhere bij takes on the value 1 in :F when i = j and the value 0 in :F when i '" j.

1 eventually became and remained zero, then the heat in the system would tend to a uniform distribution due to the circulating flow. Hence the water in each tank would approach the same limiting temperature Too = _1_ [x? 5) as t - 00, in which e is the specific heat of water in the compatible units of calories per gallon per degree. Suppose that the intended purpose of the system were to drive the temperature of the water in the tanks to some prescribed value Too' If the initial temperatures were measured, the values of x?

1 but with A now regarded as a matrix over the complex field V. 5) but now factors into a product of linear polynomials over V, PA(x) = (X - 4 ) [ ( ~- 2)’ = (x - 4)[x - (2 + 161 + 4i)] [x - (2 - 4i)l. 8) Hence the eigenvalues of A are I = 4, 2 f 4i. 10) 42 2. 11) which also has dimension 1. 1 If A is a real n x n matrix with a complex eigenvalue A. = a + ico, the associated eigenvectors can be computed by real arithmetic. Let v = x + iy. 16) Y = (Ilw)(rxI - A)x. 16), doing only real arithmetic.