# 2.1 - Math 2940 Solutions Fall 2011 Section 2.1 3 Certainly...

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Math 2940 Solutions, Fall 2011 Section 2 . 1 3 ) Certainly the planes are changed. The last two have diﬀerent equations after the addition. The new matrix is 1 0 0 1 1 0 0 0 1 so the the ﬁrst column has changed. Clearly the coeﬃcient matrix has changed. The solutions are still the same, namely x = 2, y = 3 z = 4. 5 ) If ( x, y, z ) satisﬁes the ﬁrst two equations, it automatically satisﬁes the third equation. To ﬁnd solutions we solve the ﬁrst two equations. Adding - 1 times the ﬁrst to the second gives y = 1. Plugging back into the ﬁrst gives x + z = 1. So the typical solution is ( t, 1 , 1 - t ) where t can be any number. just choose three diﬀerent values for t . 9 ) (a) 1 2 4 - 2 3 1 4 1 2 2 2 3 = 1 · 2 + 2 · 2 + 4 · 3 - 2 · 2 + 3 · 2 + 1 · 3 - 4 · 2 + 1 · 2 + 2 · 3 = 18 5 0 (b) 2 1 0 0 1 2 1 0 0 1 2 1 0 0 1 2 1 1 1 2 = 2 · 1 + 1 · 1 + 0 · 1 + 0 · 2 1 · 1 + 2 · 1 + 1 · 1 + 0 · 2 0 · 1 + 1 · 1 + 2 · 1 + 1 · 2 0 · 1 + 0 · 1 + 1 · 1 + 2 · 2 = 3 4 5 5 12 ) The products are z y x , 0 0 0 , 3 3 6 . 13 )(a) A matrix with m rows and

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2.1 - Math 2940 Solutions Fall 2011 Section 2.1 3 Certainly...

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