NYJC 2026 FM TP - Linear Algebra Set 4

1. Let 2 2 4 3 3 2 05 a a b a a b ab −−  − − be the augmented matrix of a linear system. Using elementary row operations, simplify the augmented matrix and hence (a) find the values of a and b for the system to have a unique solution, [2] (b) give a geometrical interpretation for the system of equations when 0 and 5.ab =− [2] 2. In a four-pole electrical network, the output quantities 11 and EI are given in terms of the input quantities 22 and EI by 12 12 , where , , and are non-zero.EE ab a b c dII cd    =       Denoting by , ab cd   A show that 11 22 1 1 EI a EI c d  −   =     −     A . [2] Similarly, show that 11 22 1 1 EI IE d      =         A , where and  are constants to be determined in terms of a, b, c and d. [3] Given A is orthogonal, that is T =AA I , and 1,=A b and d are non-zero, find 12 and EE in terms of 12, , and .b d I I [5]