GATE IT 2008 | Question: 29

Question

If $M$ is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct?

S1: Each row of $M$ can be represented as a linear combination of the other rows
S2: Each column of $M$ can be represented as a linear combination of the other columns
S3: $MX = 0$ has a nontrivial solution
S4: $M$ has an inverse

• A. $S3$ and $S2$
• B. $S1$ and $S4$
• C. $S1$ and $S3$
• D. $S1, S2$ and $S3$

Comments

@Sachin Mittal 1 sir thank youu soo much for confermation.

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@Sachin Mittal 1 sir plz elliminate happy mittal sir answer is it just copy paste    of madeeasy pyq book which is wrong…..

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Actually i think option C is correct only, AX=0 has a non-trivial solution if only if A is a  singular matrix.Remaining options are wrong, they might be correct if any one of the vectors can be represented as linear combinations of others.

Answer 1

Answer : None of the options are correct  ( Only statement 3 always correct )

Det ( M ) = 0 => Both columns & rows of square matrix M are Linearly Dependent.

Statement( 1 & 2 ) : A set of vectors are Linearly Dependent does’t mean that each one of them can be represented as linear combinations of other vectors. eg: 

0   0   0

0   1   2

1   0   0       // This matrix obviously have determinant 0, so columns and rows are linearly dependent, but 3^rd row ( 1   0   0  )  and first column (  [ 0   0   1 ]T ) cannot be represented as linear combinations of other rows/columns. So both S1 and S2 false.

Statement( 3 ) : This is correct, Ax = 0 will have a non trivial solution when the set of column vectors is Linearly dependent.

Statement( 4 ) : This is obviously false, A matrix have an inverse iff Det( M ) != 0

Answer 2

A matrix with number of rows equal to number of columns is square matrix and when determinant of square matrix is zero, it is called singular matrix or non- invertible matrix.

Properties of singular matrix:
Singular matrices are those where some rows or columns can be expressed by a linear combination of others.
here M is a square matrix with zero determinant i.e. M is singular |M|=0
Statements(s1,s2): correct
BY using property of singular matrix, we can see that columns or rows do not contain additional information.They are redundant and using row elimination or column elimination, matrix determinant is equal to zero.so it can be represented as linear combinations.
Statement (s3): correct
As |M| is equal to zero,it will give non trivial solution. as matrix properties say, for a non trivial solution, determinant should be equal to zero.
Statement(s4): incorrect
We know that the formula for finding the inverse of a square matrix M is: M −1 = adjoint(M)/|M|
If |M| = 0, then M −1 would given an indeterminate form; i.e. its inverse will not exist.

Answer 3

only S3 is true