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Each row of M can be represented as a linear combination of the other rows

1)Does that mean linear combination of other rows will be 0?  how ?

2)And also , is linear combination means add, subtract, multiply and divide , but not squaring or root or exponential operation,right?

https://gateoverflow.in/3319/gate2008-it-29

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1) If each row of M can represented as the linear combination of the other rows 

then linear combination of all the rows (not other rows) will be zero.

Reason

Since each row of M can be represented as the linear combination of the other rows so we have

R1 = K2R2 + K3R3 + K4R4 + ............+KnRn

=> K2R2 + K3R3 + K4R4 + ............+KnRn - R1 = 0

=> K1R1 + K2R2 + K3R3 + K4R4 + ............+KnRn = 0 ( here k1 = -1 )

So, the linear combination of all the rows is zero.

2) Yes, linear combination means only add,substract,multiply and divide not any other operation.

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4 Comments

Look at this

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Note that in my above comment

R1 is written as the linear combination of R2 and R3

R2 is written as the linear combination of R1 and R3

R3 is written as linear combination of R1 and R2

Here R1 + 3R2 - R3 = 0

So,the linear combination of all the rows are zero.
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@

Just a converse of this 

If All rows of matrix are written as linear combination which is equal to zero ====> Each row can be written as linear combinarion of other rows

This is false right ?? ,,if in linear combination of all rows if some constants are zero then ..it might not be possible to write each row as linear combination of other ,,,

but 

If each row can be written as linear combinarion of other rows ====> All rows of matrix are written as linear combination which is equal to zero

THis is true as u explained above

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