Let A = left[ {begin{array}{*{20}{c}}1&23&4end{array}} right] and B = left[ {begin{array}{*{

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

easy

Let $A = \left[ {\begin{array}{{20}{c}}1&2\\3&4\end{array}} \right]$ and $B = \left[ {\begin{array}{{20}{c}}a&0\\0&b\end{array}} \right]\;,a,b \in N$ then

there cannot exist any $B$ such that $AB = BA$

there exist more then one but finite number of $B's$ such that $AB=BA$

there exists exactly one $B$ such that $AB = BA$

there exist infinitely many $B's$ such that $AB = BA$

(AIEEE-2006)

Solution

$A=\left[\begin{array}{ll}{1} & {2} \\ {3} & {4}\end{array}\right] \quad $ $B=\left[\begin{array}{ll}{a} & {0} \\ {0} & {b}\end{array}\right]$

$A B=\left[\begin{array}{ll}{a} & {2 b} \\ {3 a} & {4 b}\end{array}\right]$

$B A=\left[\begin{array}{ll}{a} & {0} \\ {0} & {b}\end{array}\right]\left[\begin{array}{ll}{1} & {2} \\ {3} & {4}\end{array}\right]$$=\left[\begin{array}{ll}{a} & {2 a} \\ {3 b} & {4 b}\end{array}\right]$

Hence, $A B=B A$ only when $a=b$

$\therefore$ There can be infinitely many $B^{\prime} s$

for which $A B=B A$

Standard 12

Mathematics

Let $P$ be a square matrix such that $P ^2= I – P$. For $\alpha, \beta, \gamma, \delta \in N$, if $P ^\alpha+ P ^\beta=\gamma I -29 P$ and $P ^\alpha- P ^\beta=$ $\delta I-13 P$, then $\alpha+\beta+\gamma-\delta$ is equal to $……..$.

normal

(JEE MAIN-2023)

If $A =$ $\left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)$ satisfies the equation $x^2 – (a + d) x + k = 0$, then

medium

$A=\left[\begin{array}{l}a_{i j}\end{array}\right]_{m\times n}$ is a square matrix, if

easy

If $A = \left[ {\begin{array}{{20}{c}}
1&0&0 \\
2&1&0 \\
{ – 3}&2&1
\end{array}} \right]\,$ and $B = \left[ {\begin{array}{{20}{c}}
1&0&0 \\
{ – 2}&1&0 \\
7&{ – 2}&1
\end{array}} \right]$ then $AB$ equals

hard

(AIEEE-2012)

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^2=3 A +\alpha I$. If $A ^4=21 A +\beta I$, then

hard

(JEE MAIN-2023)

Let $A = \left[ {\begin{array}{*{20}{c}}1&2\\3&4\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}a&0\\0&b\end{array}} \right]\;,a,b \in N$ then