GATE | Gate IT 2008 | Question 28

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


Answer: (D)

Explanation:

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
Let us understand this concept in detail.
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.
Note: this comes into the basic fundamental of matrix.so read basics.

This solution is contributed by Nitika Bansal.

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