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jonathan's question at Yahoo! Answers ( span ( H ) )

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello jonathan,

Clearly
$$\mbox{rank }\begin{bmatrix}{2}&{-1}&{\;\:3}& \;\;2\\{0}&{\;\;2}&{-2}&-2\\{0}&{\;\;0}&{\;\;3}&\;\;3\end{bmatrix}=3$$

Then, $(2,-3,4,a)$ belongs to $H$ if and only if:

$$\mbox{rank }\begin{bmatrix}{2}&{-3}&{\;\:4}& \;\;a\\{2}&{-1}&{\;\:3}& \;\;2\\{0}&{\;\;2}&{-2}&-2\\{0}&{\;\;0}&{\;\;3}&\;\;3\end{bmatrix}=3$$
Equivalently, if and only if
$$\det\;\begin{bmatrix}{2}&{-3}&{\;\:4}& \;\;a\\{2}&{-1}&{\;\:3}& \;\;2\\{0}&{\;\;2}&{-2}&-2\\{0}&{\;\;0}&{\;\;3}&\;\;3\end{bmatrix}=0$$

Easily we verify that the previous determinant is $12(9-a)$ so,

$(2,-3,4,a)$ belongs to $H$ if and only if $a=9$