If A = left[ {begin{array}{*{20}{c}}1&2{ - 3}&0end{array}} right] and B = left[ {begin{array

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

easy

If $A = \left[ {\begin{array}{{20}{c}}1&2\\{ - 3}&0\end{array}} \right]$ and $B = \left[ {\begin{array}{{20}{c}}{ - 1}&0\\2&3\end{array}} \right]$, then

${A^2} = A$

${B^2} = B$

$AB \ne BA$

$AB = BA$

Solution

(c) Since ${A^2} = \left[ {\begin{array}{*{20}{c}}1&2\\{ – 3}&0\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&2\\{ – 3}&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ – 4}&2\\{ – 3}&{ – 6}\end{array}} \right] \ne A$

${B^2} = \left[ {\begin{array}{*{20}{c}}{ – 1}&0\\2&3\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}{ – 1}&0\\2&3\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\4&9\end{array}} \right] \ne B$

Now $AB = \left[ {\begin{array}{*{20}{c}}1&2\\{ – 3}&0\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}{ – 1}&0\\2&3\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3&6\\3&0\end{array}} \right]$

and $BA = \left[ {\begin{array}{*{20}{c}}{ – 1}&0\\2&3\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&2\\{ – 3}&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ – 1}&{ – 2}\\{ – 7}&4\end{array}} \right]$

Obviously, $AB \ne BA$.

Standard 12

Mathematics

If $A = \left[ {\begin{array}{*{20}{c}}\lambda &1\\{ – 1}&{ – \lambda }\end{array}} \right]$, then for what value of $\lambda ,\,{A^2} = O$

easy

Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is……………………

hard

(JEE MAIN-2024)

Let $A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$. Let $\alpha, \beta \in R$ be such that $\alpha A^{2}+\beta A=2 I$. Then $\alpha+\beta$ is equal to –

medium

(JEE MAIN-2022)

If $A$ and $B$ are $3 \times 3$matrices such that $AB = A$ and $BA = B$, then

medium

If $I$ is a unit matrix, then $3I$ will be

normal

If $A = \left[ {\begin{array}{*{20}{c}}1&2\\{ - 3}&0\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\2&3\end{array}} \right]$, then