Question

Let A and B be n x n matrices.

The determinant of A is the product of the diagonal entries in A. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The statement is true because the determinant of any triangular matrix A is the product of the entries on the main diagonal of A.

B.

The statement is false because the determinant of the 2×2 matrix A = __ is not equal to the product of the entries on the main diagonal of A.

​(Type an integer or simplified fraction for each matrix​element.)

C.

The statement is true because the determinant of any square matrix A is the product of the entries on the main diagonal of A.

Answer 1

The statement is FALSE, option B is correct

Explanation:

What option A says is true, the determinant of any triangular matrix A is the product of the entries on the main diagonal of A.

However, it is not stated that A is triangular, so this afirmation is not enough to prove that det(A) is the product of the diagonal elements in A. So we cant count on option A.

Option C is not valid, and the argument is based on a wrong claim. The product of the entries of the main diagonal of a matrix A isnt necessarily det(A). However, the claim is true when A is traingular, as option A states.

Option B is the correct one, the 2x2 matrix
`A = \left[\begin{array}{cc}1&1\\1&1\end{array}\right] ,` has determinant equal to 0, because it has 2 equal rows. However the product of the elements of the diagonal gives 1, so the product of the entries of the diagonal of A isnt equal to det(A).