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:

(B) and (D) only

The correct answer is Option (2) → (B) and (D) only

Given matrix:

$\begin{bmatrix} a-b & 0 & 0 \\ 0 & b-c & 0 \\ 0 & 0 & c-2 \end{bmatrix}$ is a scalar matrix.

For a scalar matrix, all diagonal elements are equal:

$a-b = b-c = c-2 = k$ (some constant)

Also, $a+b+c=0$

From $a-b = b-c \Rightarrow a-b = b-c \Rightarrow a - 2b + c = 0$

Also, $a+b+c = 0$

Subtract the two equations: $(a+b+c) - (a-2b+c) = 0 - 0 \Rightarrow 3b = 0 \Rightarrow b = 0$ ✅

Then, $a+b+c = a+0+c=0 \Rightarrow a + c = 0 \Rightarrow a=-c$

Also, $b-c = 0-c=-c = k$ and $c-2 = k \Rightarrow -c = c-2 \Rightarrow 2c=2 \Rightarrow c=1$ ✅

Then, $a=-c=-1$

Check $a-b = -1-0=-1$, $b-c=0-1=-1$, $c-2=1-2=-1$ ✅ All equal, matrix is scalar.

True statements: (B) b=0, (D) c=1