Question

A and B are n×n matrices. Check the true statements below: A. (detA)(detB)=detAB. B. The determinant of A is the product of the diagonal entries in A. C. If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. D. An elementary row operation on A does not change the determinant.

Answer 1

Answer with Step-by-step explanation:

We are given that

A and B are matrix.

A.We know that for two square matrix A and B

Then, $det(AB)=det(A)\cdot det(B)$

Therefore, it is true.

B. det A  is  the product of diagonal entries in A.

It is not true for all matrix.It is true for upper triangular matrix.

Hence, it is false.

C.$\lambda+5=0$

$\lambda=-5$

When  is a factor of the characteristics polynomial of A then -5 is an eigenvalue of A not 5.

Hence, it is false.

D.An elementary row operation on A does not change the determinant.

It is true because when an elementary operation applied then the value of matrix A does not change.