CMI-2021 Data Science

Question

Which of the following statement(s) is/are true for an arbitrary n × n matrix A?

(a) Exchanging two rows of A does not change its determinant.

(b) Exchanging two rows of A does not change its trace.

(c) Replacing each diagonal element of A with a 1 does not change its determinant.

(d) Exchanging two columns of A negates its determinant.

Answer 1

(a) : exchanging two rows will bring a negative sign in the determinant hence changing it. So, false.

(b) : exchanging two rows will change the diagonal elements hence the trace will be changed. So, false.

(c ): by taking a small matrix example, one can easily prove that by changing the diagonal elements to 1, the determinant will change. So, false.

(d) : True.