NDA 2019 Maths · Mathematics · Matrices · medium

If A and B are 2×2 matrices with AB=O, then A=O or B=O. 2. For square matrices of the same order, |A+B|=|A|+|B|.

Single correct option

Consider the following statements:
1. If

A

and

B

are

2\times2

matrices with

AB=O

, then

A=O

B=O

.
2. For square matrices of the same order,

|A+B|=|A|+|B|

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (D) Neither 1 nor 2

Statement 1 is false: with

A=\begin{pmatrix}1&0\\0&0\end{pmatrix}

and

B=\begin{pmatrix}0&0\\0&1\end{pmatrix}

we get

AB=O

although neither is a zero matrix. Statement 2 is false: with

A=B=I_2

|A+B|=4

but

|A|+|B|=2

. The determinant is multiplicative, not additive. So neither statement is correct.

Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake