Given that ( c 1 , c 2 ,c 3 ) is a solution to the following homogeneous systemo

Question

Given that ( c1, c2 ,c3 ) is a solution to the following homogeneous systemof equations:
a 11 x 1 + a 12 x2 + a 13 x 3 = 0 a 21 x 1 + a 22 x2 + a 23 x 3 = 0 a 31 x 1 + a 32 x2 + a 33 x 3 = 0 Demonstrate that for any real number k, (kc 1 , kc2 , kc 3) is also a solution. I really have no idea what to do with this. I started bysubstituting the c1 in for x1 and so on, butI wasn't sure if that was correct, and then I didn't know whatto do next. Any help would be appreciated!! Given that ( c1, c2 ,c3 ) is a solution to the following homogeneous systemof equations:
a 11 x 1 + a 12 x2 + a 13 x 3 = 0 a 21 x 1 + a 22 x2 + a 23 x 3 = 0 a 31 x 1 + a 32 x2 + a 33 x 3 = 0 Demonstrate that for any real number k, (kc 1 , kc2 , kc 3) is also a solution. I really have no idea what to do with this. I started bysubstituting the c1 in for x1 and so on, butI wasn't sure if that was correct, and then I didn't know whatto do next. Any help would be appreciated!!

Explanation / Answer

thats a good start. we know a11 c1 + a12 c2 + a13 x3 =0, because (c1, c2,c3) is asolution. now try (kc1, kc2, kc3). a11 * k * c1 + a12 *k * c2 + a13 *k * x3 = k (a11 c1 + a12 c2 + a13x3) = k * 0 =0, so these values also satisfy the firstequation. Repeat for the second two equations and you're done!

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