GATE CSE 2024 | Set 1 | Question: 2

Question

The product of all eigenvalues of the matrix $\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ is

• A. $-1$
• B. $0$
• C. $1$
• D. $2$ 

Answer 1

Product of eigen values is Determinant of the matrix.
Since, Columns are Linearly Dependent (C₃ = 2*C₂ - C₁) we can say Determinant will be zero.

(B) is correct option.
 

Answer 2

Product of eigen values is the determinant of the matrix.

Here the rows are linearly dependent as follows:

row3 = 2*row2 - row1

Thus, the determinant is zero.

Answer: option B

Answer 3

All columns of the matrix are Linearly Dependant on each other, so the determinant is 0.

Hence the correct answer is B.

Answer 4

B)0