Question

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

A. det (A) = 0 or det (B) = 0 

B. det (A) + det (B) = 0 

C. det (A) = 0 and det (B) = 0 

D. A + B = 0

Answer 1

D. A + B = 0

We are given that, 

Matrices A and B are square matrices. 

Order of matrix A = 2 

Order of matrix B = 2 

Det (A + B) = 0 

We need to find the condition at which det (A + B) = 0. 

Let, 

Matrix A = [a_ij] Matrix B = [b_ij] 

Since their orders are same, 

We can express matrices A and B as,

A + B = [a_ij + b_ij] 

⇒ |A + B| = |a_ij + b_ij| …(i) 

Also, 

We know that,

Det (A + B) = 0 

That is, 

|A + B| = 0 

From (i), 

|a_ij + b_ij| = 0 

If 

⇒ [a_ij + b_ij] = 0 

Each corresponding element is 0. 

⇒ A + B = 0

Thus, 

det (A + B) = 0 is possible when A + B = 0.