If \(\begin{bmatrix}a-b&0&0\\0&b-c&0\\0&0&c-2\end{bmatrix}\) is a scalar matrix such that \(a + b + c = 0\), then, which of the following are TRUE?
(A) \(a = 0\)
(B) \(b = 0\)
(C) \(a = 1\)
(D) \(c = 1\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
In a scalar matrix all the diagonal entries are equal:
\(a - b = b - c = c - 2\)
From the first equality:
\(a + c = 2b\)
Putting this in \(a + b + c = 0\):
\(3b = 0, \quad b = 0\)
From \(b - c = c - 2\) with b = 0:
\(-c = c - 2, \quad c = 1\)
\(a = -b - c = -1\)
So a = −1, b = 0 and c = 1 (every diagonal entry is −1).
Checking the statements
(A) \(a = 0\) — Incorrect. a is −1.
(B) \(b = 0\) — Correct. \(3b = 0\).
(C) \(a = 1\) — Incorrect. a is −1.
(D) \(c = 1\) — Correct. \(-c = c - 2\) gives c = 1.