Compute Row Space Of 2-By-3 Matrix

Level 2

Consider the matrix

B = ( 8 − 5 2 0 2 1 ) . B= \begin{pmatrix} 8 & -5 & 2 \\0 & 2 & 1 \end{pmatrix}.

Note that the column space C ( B ) C(B) is just R 2 \mathbb{R}^2 , since the first two columns are linearly independent. The row space R ( B ) R(B) has equation a x + b y + c z = 0 ax+by+cz = 0 , where a , b , c ∈ Z , a > 0 , a,b,c\in \mathbb{Z}, a>0, and gcd ⁡ ( a , b , c ) = 1 \gcd(a,b,c) =1 . What is a + b + c ? a+b+c?


The answer is 1.

Sameer Kailasa by Sameer Kailasa , 25, USA

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1 solution

We have two equations and three unknowns i.e. 8a - 5b + 2c = 0 and 2b + c = 0.
But the additional constraints of a,b,c being integers, a > 0 and gcd(a,b,c) = 1 lets us solve the equations as follows: 1) Substitute c = -2b in the first equation 2) We get b = (8/9)*a. 3) If we let a = 9 (satisfies the constraints), we get b = 8 and c = -16. 4) The gcd(a,b,c) is still 1. So a + b + c = 9 + 8 -16 = 1.

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