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

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3 and 4 .Determinants and Matrices

easy

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

$( - 2,\, - 2)$

$(2,\, - 2)$

$( - 2,\,2)$

$(2,\,2)$

Solution

(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 $ $a = – 2$, $b = – 2$.

Standard 12

Mathematics

If $A = \left[ {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right]$, then ${A^4}$ is equal to

easy

Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$

Men workers
Women workers

$I$ $30$ $25$

$II$ $25$ $31$

$III$ $27$ $26$

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

easy

$A$ and $B$ are two non- singular square matrices of each $3 \times 3$ such that $AB = A$ and $\left| {A + B} \right| \ne 0$, then

normal

Matrix theory was introduced by

normal

Let $S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}$ and let $T_{n}=\left\{A \in S: A^{n(n+1)}=I\right\}$. Then the number of elements in $\bigcap \limits_{n=1}^{100} T_{n}$ is

hard

(JEE MAIN-2022)

	Men workers	Women workers
$I$	$30$	$25$
$II$	$25$	$31$
$III$	$27$	$26$

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

Solution

Similar Questions

If $A = \left[ {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right]$, then ${A^4}$ is equal to

$A$ and $B$ are two non- singular square matrices of each $3 \times 3$ such that $AB = A$ and $\left| {A + B} \right| \ne 0$, then

Matrix theory was introduced by

Let $S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}$ and let $T_{n}=\left\{A \in S: A^{n(n+1)}=I\right\}$. Then the number of elements in $\bigcap \limits_{n=1}^{100} T_{n}$ is

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