Question

If A and B are any $ 2 \times 2 $ matrix, then $ |A + B| = 0 $ implies
A. $ |A| + |B| = 0 $
B. $ |A| = 0 $ or $ |B| = 0 $
C. $ |A| = 0 $ and $ |B| = 0 $
D. None of these

Answer 1

Hint: Here, in this question we are given a $ 2 \times 2 $ matrix and we are supposed to find the determinant of A+B. Here, the zero also indicates a matrix, which is none other than, zero matrix. In order to find the solution of the given question, we will use the concept of determinants. First, we will find out the value of A in terms of B. Next, we will take the determinant of the obtained value and get our required answer.

Complete step by step solution:
Given A and B are $ 2 \times 2 $ matrices and the determinant of (A+B) is equal to zero. That is,
 $ |A + B| = 0 $
Here, zero indicates zero matrix.
According to the question, we get,
 $
   \Rightarrow A + B = 0 \\
   \Rightarrow A = 0 - B \\
   \Rightarrow A = - B \;
  $
As, now we got the value of A in terms of B, we will take determinant on both sides of the equation.
 $ \Rightarrow |A| = - |B| $
To further simplify we take the determinant of B on the left-hand side of the equation and we get,
 $ \Rightarrow |A| + |B| = 0 $
Therefore, we get determinant of A + determinant of B is equal to zero, i.e., $ |A| + |B| = 0 $
Hence, the correct option is (A).
So, the correct answer is “Option A”.

Note: The given question was very easy to solve if you know the concept of determinants. The common mistake which almost all the students tend to make is to assume that determinant and matrix are the same. The symbols of matrix and determinant are different from each other. Here, we just used the concept of determinant and basic addition and subtraction.