Can a matrix have a linear non dependent rows even if the determinant is 0

Course Q&A Math for Machine Learning Linear Algebra for Machine Learning and Data Sc...
1

in the following matrix the determinant is 0 but it does not seem to have linear dependency.

1 1 1
0 2 2
0 0 0

Is my understanding correct ?
or this is linear dependent as I can multiply 1st or second row with 0 which will give me 3rd row ? is multiplying with 0 to identify linear dependency a correct approach ?

2

Multiplying by 0 is never useful for reduction: it just kills everything.

There is more than one way for a matrix to be singular. In this case notice that the matrix has rank 2, which makes it singular, even if the first two rows are linearly independent.

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You are right.
ā€˜Being linearly dependentā€™ means that you can get a row or column of a matrix by linear combination of all other rows or columns.
As you indicated, you can get the 3rd row from the linear combination of the 1st and 2nd row where of course the coefficients are 0 and 0.
For that reason, you can easily deduct that if a matrix has an all-zero row or column, the matrix is automatically singular.

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5

Thanks @paulinpaloalto , @taekyo_lee for the explanation. it cleared my doubt.