Question

show that A has rank 2 if and only if one or more of the determinants is non-zero. see attachment. Any one helps me, please

Answer 1

\[A = \left[\begin{matrix}a11 & a12 & a13 \\ a21 & a22 & a23\end{matrix}\right]\]
the determinants are \[\left[\begin{matrix}a11 & a12 \\ a21 & a22\end{matrix}\right]\];\[\left[\begin{matrix}a11 & a13 \\ a21 & a23\end{matrix}\right]\] and \[\left[\begin{matrix}a12 & a13 \\ a22 & a23\end{matrix}\right]\]

Answer 2

@myko any idea, please

Answer 3

I think this is true by definition. The determinants you wrote are the so called minors of A. Each of them is a 2x2 matrix. If any of them is not 0 then A has Rank 2

Answer 4

but how to put it into logic? the definition of rank A

Answer 5

Rank is the size of the biggest minor of A with non 0 determinant

Answer 6

Thanks a lot. I 'll try to do with that

Answer 7

or to say it other way the number of linearly independent columns (raws)

Answer 8

and it's true with the particular case like mine. because we cannot have determinant of 3X5 matrix, right?

Answer 9

if any of thoes determinants is not 0 it means that the corresponding columns in A matrix are independent

Answer 10

Got it. thank you

Answer 11

ok. Yw