16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 1.3.8

Systems of Linear Equations

Mathematics — Linear Algebra · homogeneous-systems-2x2

Problem

Consider the system of equations AX=0AX = 0 where A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} is a 2×22 \times 2 matrix over the field FF. Prove the following. (a) If every entry of AA is 0, then every pair (x1,x2)(x_1, x_2) is a solution of AX=0AX = 0. (b) If ad−bc≠0ad - bc \neq 0, the system AX=0AX = 0 has only the trivial solution x1=x2=0x_1 = x_2 = 0. (c) If ad−bc=0ad - bc = 0 and some entry of AA is different from 0, then there is a solution (x10,x20)(x_1^0, x_2^0) such that (x1,x2)(x_1, x_2) is a solution if and only if there is a scalar yy such that x1=yx10x_1 = yx_1^0, x2=yx20x_2 = yx_2^0.

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