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solq5 - Solutions to quiz 5(October 7 1 Writing X = 13 31 a...

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Solutions to quiz # 5 (October 7) 1. Writing X = bracketleftbigg a b c d bracketrightbigg , we obtain bracketleftbigg 1 3 3 1 bracketrightbigg bracketleftbigg a b c d bracketrightbigg = bracketleftbigg a b c d bracketrightbigg bracketleftbigg 1 3 3 1 bracketrightbigg ⇐⇒ bracketleftbigg a + 3 c b + 3 d 3 a + c 3 b + d bracketrightbigg = bracketleftbigg a + 3 b 3 a + b c + 3 d 3 c + d bracketrightbigg ⇐⇒ a + 3 c = a + 3 b b + 3 d = 3 a + b 3 a + c = c + 3 d 3 b + d = 3 c + d ⇐⇒ b = c a = d. Answer: X = bracketleftbigg a b b a bracketrightbigg , where a and b can be any numbers. 2. To find a basis of the kernel, we solve the system of linear equations: x 1 x 2 x 3 x 4 1 2 0 3 | 0 1 2 1 2 | 0 1 2 - 1 4 | 0 -→ 1 2 0 3 | 0 0 0 1 - 1 | 0 0 0 - 1 1 | 0 -→ 1 2 0 3 | 0 0 0 1 - 1 | 0 0 0 0 0 | 0 , from which the kernel consists of the vectors x 1 = - 2 t 2 - 3 t 1 x 2 = t 2 x 3 = t 1 x 4 = t 1 = t 1 - 3 0 1 1 + t 2 - 2 1 0 0 , where t 1 and t 2 can be any numbers. Hence vectors
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