Question

If A, B be real 2×2 matrices such that AA^T+BB^T =0,can A and B be invertible

Answer 1

$\textbf{Given:}$

$\text{A and B are real 2{\times}2 matrices and A\,A^T+B\,B^T=0 }$

$\textbf{To find:}$

$\text{Can A and B be invertible?}$

$\textbf{Solution:}$

$\text{Consider,}$

$A\,A^T+B\,B^T=0$

$\implies\,A\,A^T=-B\,B^T$

$\text{Taking determinant on bothsides}$

$\implies\,|A\,A^T|=|-B\,B^T|$

$\implies\,|A|\,|A^T|=|-B|\,|B^T|$

$\text{We know that,}$

$\textbf{If A is a square matrix of order n, then}$

$\boxed{\bf|kA|=k^n|A|}$

$\boxed{\bf|A^T|=|A|}$

$\implies\,|A|\,|A|=(-1)^2|B|\,|B|$

$\implies\,|A|^2=|B|^2$

$\implies\dfrac{|A|^2}{|B|^2}=1$

$\text{Taking square root on bothsides}$

$\implies\dfrac{|A|}{|B|}{\neq}0$

$\implies\,|A|{\neq}0\;\text{and}\;|B|{\neq}0$

$\therefore\text{A and B are invertible}$

$\textbf{Answer:}$

$\textbf{A and B are invertible matrices}$

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