If A = [ * 20 c 2&2 &b ] and A^2 = O, then (a,b) = - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}2&2\\a&b\end{array}} \right]$ and ${A^2} = O$, then $(a,b) = $

✓
$( - 2,\, - 2)$
B
$(2,\, - 2)$
C
$( - 2,\,2)$
D
$(2,\,2)$

✓

Answer

Correct option: A.

$( - 2,\, - 2)$

a
(a) ${A^2} = \left[ {\begin{array}{*{20}{c}}2&2\\a&b\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}2&2\\a&b\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{4 + 2a}&{4 + 2b}\\{2a + ab}&{2a + {b^2}}\end{array}} \right] = 0 = \left[ {\begin{array}{*{20}{c}}0&0\\0&0\end{array}} \right]$

$ \Rightarrow \,\,4 + 2a = 0,4 + 2b = 0,$$2a + ab = 0,$

$2a + {b^2} = 0$ must be consistent.

$ \Rightarrow $ $a = - 2$, $b = - 2$.

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