An even number is the determinant of
(A) \(\begin{bmatrix}1 & -1\\-1 & 5\end{bmatrix}\) (B) \(\begin{bmatrix}13 & -1\\-1 & 15\end{bmatrix}\) (C) \(\begin{bmatrix}16 & -1\\-11 & 15\end{bmatrix}\) (D) \(\begin{bmatrix}6 & -12\\11 & 15\end{bmatrix}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
For a 2 × 2 matrix, the determinant is ad − bc:
\((A):\ 1 \times 5 - (-1)(-1) = 5 - 1 = 4\)
\((B):\ 13 \times 15 - (-1)(-1) = 195 - 1 = 194\)
\((C):\ 16 \times 15 - (-1)(-11) = 240 - 11 = 229\)
\((D):\ 6 \times 15 - (-12)(11) = 90 + 132 = 222\)
So the determinants are 4, 194, 229 and 222.
Checking the statements
(A) \(\begin{bmatrix}1 & -1\\-1 & 5\end{bmatrix}\) — Correct. Its determinant 4 is even.
(B) \(\begin{bmatrix}13 & -1\\-1 & 15\end{bmatrix}\) — Correct. Its determinant 194 is even.
(C) \(\begin{bmatrix}16 & -1\\-11 & 15\end{bmatrix}\) — Incorrect. Its determinant 229 is odd.
(D) \(\begin{bmatrix}6 & -12\\11 & 15\end{bmatrix}\) — Correct. Its determinant 222 is even.