Question

If A and B are square matrices of order 2, then det (A + B) = 0 is possible only when

Options

• det (A) = 0 or det (B) = 0

• det (A) + det (B) = 0

• det (A) = 0 and det (B) = 0

•  A + B = O

Answer 1

(d) A + B = O

\[\text{ Let }A = \left[ a_{i j} \right]\text{ and }B = \left[ b_{i j} \right]\text{ be a square matrix of order 2 .} \]
\[\text{ As their orders are same,  A + B is defined as}\]
\[A + B = \left[ a_{i j} + b_{i j} \right]\]
\[ \Rightarrow \left| A + B \right| = \left| a_{i j} + b_{i j} \right|\]
Now, 
\[\left| A + B \right| = 0\]
\[ \Rightarrow \left| a_{i j} + b_{i j} \right| = 0\]
\[ \Rightarrow \left[ a_{i j} + b_{i j} \right] = 0 \left[\text{ each corrsponding term is 0 }\right]\]
\[ \Rightarrow A + B = 0\]