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JEE Main 202229 JulyMorning Shift
Exam
JEE Main
Year
2022
Date
29 July
Shift
Morning Shift
Subject
Mathematics
Chapter
Matrices

JEE Main 2022 · 29 July · Morning Shift · Mathematics · Matrices

Let A and B be two 3 × 3 non-zero real matrices such that AB is a...

Let AA and BB be two 3×33 \times 3 non-zero real matrices such that ABAB is a zero matrix. ThenLet A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then

Options

  1. AThe system of linear equations AX=0AX = 0 has a unique solution
  2. BThe system of linear equations AX=0AX = 0 has infinitely many solutions
  3. CBB is an invertible matrix
  4. Dadj(A)\left(A\right) is an invertible matrix

Correct answer

B. The system of linear equations AX=0AX = 0 has infinitely many solutions

Step-by-step solution

Given,

AB=0⇒∣AB∣=0AB = 0 ⇒ \left|AB\right| = 0

Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then

If ∣A∣≠0,B=0\left|A\right| \ne 0, B = 0 (not possible)

If \left|B\right| \ne 0, A = 0(notpossible)</p><p>Hence∣A∣=∣B∣=0(not possible)</p><p>Hence \left|A\right| = \left|B\right| = 0

So, if ∣A∣=∣B∣=0\left|A\right| = \left|B\right| = 0 then A \text{ and } B are non-invertible matrix,

So, AX = 0 has infinitely many solutions

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