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Let \mathrm{A\: and\: B} be two \mathrm{3 \times 3} non-zero real matrices such that \mathrm{A B} is a zero matrix. Then

Option: 1

the system of linear equations \mathrm{AX}=0 has a unique solution
 


Option: 2

the system of linear equations \mathrm{AX}=0 has infinitely many solutions
 


Option: 3

\mathrm{B} is an invertible matrix
 


Option: 4

\operatorname{adj}(\mathrm{A}) is an invertible matrix


Answers (1)

best_answer

                \mathrm{A B=0 \Rightarrow|A B|=0}\\ \mathrm{|A||B|=0}\\

                

\mathrm{|A|=0,                \mathrm{|B|=0    

\mathrm{if |A| \neq 0, B=0}\text{ (not possible)}\\ \mathrm{if |B| \neq 0, A=0 }\text{(not possible)}\\ \mathrm{Hence |A|=|B|=0}\\ \Rightarrow \text{Ax=0 has infinitely many solutions.}

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Rishabh

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