By Hedley G. Martin and N. Hiller (Auth.)

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1) is in- DETERMINANTS AND LINEAR EQUATIONS 39 consistent when not all the six 2 c 2 determinants corresponding with D1 and D 2 vanish, as in (5) and (6). (5) x+ y=1, 2x + 2y = 3, 3x+3y = 5. 1 1 1 E = 2 2 3 = 1(0) —3(0)± 5 (0) = 0. 3 3 5 Here E = 0, all the 2C2 minors = land all six D 1, D 2 (6) 2x-3y = 1, 6x — 9y = 3 , 4x —6y = —2. ~ O. Two coincident lines ~ S = 2 —3 1 6 —9 3 4 —6 —2 = 1(0) — 3(0) — 2(0) = 0. Here E = 0, all the 2 c 2 minors = 0 and four of the D 1, 0. When all the determinants corresponding with D 1 and D 2 vanish the system is consistent but there is an unlimited number of solutions, as in (7).

There are two respects in which matrix multiplication and its implications differ from the multiplication of numbers. 54 MATHEMATICS FOR ENGINEERING AND TECHNOLOGY If A and B are conformable for the products AB and BA, then in general AB BA, that is the law of commutation does not apply. To distinguish between these two products, in the first A is said to premultiply B whereas in the second it is said to post-multiply B. The relation AB = 0 does not imply that A = 0 or that B = O. Similarly, A2 = 0 does not imply that A is a null matrix.

C2 Obtain the derivative of D = 3 --1 1 2 ex 1 0 c DETERMINANTS AND LINEAR EQUATIONS By columns dD dx 2c 1 2 = 0 ex 1 45 2 -}- O O c c 0 2 3 ex 1 —1 0 x 1 0 3 ex 0 = (x3 +3x2 +2)ex-3. 11 1. By inspection factorise 1 1 1 s t u tu us st 2. Multiply together, in the order shown, the determinants 0 2 3 1 4 7 6 4 2 2 3 1 —1 0 2 5 3 1 and evaluate the product determinant. Verify that a change in the order does not affect the value of the product. 3. Express the square of the determinant 0 b a D= a 0 c c 0 as one of order three.