Let A=left[a_{i j}right]_{2 × 2} where a_{i j} ≠ 0 for all i, j and A^2=I . Let a be sum of all d

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3 and 4 .Determinants and Matrices

hard

Let $A=\left[a_{i j}\right]_{2 \times 2}$ where $a_{i j} \neq 0$ for all $i, j$ and $A^2=I$. Let a be the sum of all diagonal elements of $A$ and $b =| A |$, then $3 a ^2+4 b ^2$ is equal to

$7$

$14$

$3$

$4$

(JEE MAIN-2023)

Solution

$\begin{aligned}& \text { Let } A=\left[\begin{array}{ll} p & l \\r & s\end{array}\right] \\& A^2=\left[\begin{array}{ll} p ^2+ qr & pq + qs \\pr + rs & qs + s ^2\end{array}\right] \\& \Rightarrow p ^2+ qr =1(1) pq + qs =0 \Rightarrow q ( p + s )=0 \\& \Rightarrow s ^2+ qr =1(2) pr + rs =0 \Rightarrow r(p+s)=0\end{aligned}$

Equation $(1)$ – equation $(2)$

$p ^2= s ^2 \Rightarrow p + s =0$

Now $3 a^2+4 b^2$

$=3(p+s)^2+4(p s-q r)^2$

$=3.0+4\left(-p^2-q r\right)^2=4\left(p^2+q r\right)^2=4$

Standard 12

Mathematics

Slimplify $\cos \theta \left[ {\begin{array}{{20}{l}}
{\cos \theta }&{\sin \theta } \\
{ – \sin \theta }&{\cos \theta }
\end{array}} \right]$ $ + \sin \theta \left[ {\begin{array}{{20}{c}}
{\sin \theta }&{ – \cos \theta } \\
{\cos \theta }&{\sin \theta }
\end{array}} \right]$

easy

Let $p$ , $q$ , $r$ are three real numbers satisfying $\left[ {p\,\,q\,\,r} \right]\left[ {\begin{array}{*{20}{c}}
2&p&q \\
{ – 3}&q&{ – p + r} \\
{12}&r&{ – q + 3r}
\end{array}} \right] = \left[ {5\,\,\,b\,\,c} \right]$ , then minimum value of $(b + c)$ is

normal

If ${a_{ij}} = \frac{1}{2}(3i – 2j)$ and $A = {[{a_{ij}}]_{2 \times 2}}$, then $A$ is equal to

easy

If $\left[ {\begin{array}{*{20}{c}}{2 + x}&3&4\\1&{ – 1}&2\\x&1&{ – 5}\end{array}} \right]$is a singular matrix, then $x$ is

easy

Let $S=\left\{n \in N \mid\left(\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right)^{n}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \forall a, b, c, d \in R\right\}$, where $i=\sqrt{-1} .$ Then the number of $2 -$ digit numbers in the set $\mathrm{S}$ is $……$

medium

(JEE MAIN-2021)

Let $A=\left[a_{i j}\right]_{2 \times 2}$ where $a_{i j} \neq 0$ for all $i, j$ and $A^2=I$. Let a be the sum of all diagonal elements of $A$ and $b =| A |$, then $3 a ^2+4 b ^2$ is equal to

Solution

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Slimplify $\cos \theta \left[ {\begin{array}{{20}{l}}
{\cos \theta }&{\sin \theta } \\
{ – \sin \theta }&{\cos \theta }
\end{array}} \right]$ $ + \sin \theta \left[ {\begin{array}{{20}{c}}
{\sin \theta }&{ – \cos \theta } \\
{\cos \theta }&{\sin \theta }
\end{array}} \right]$

Let $p$ , $q$ , $r$ are three real numbers satisfying $\left[ {p\,\,q\,\,r} \right]\left[ {\begin{array}{*{20}{c}}
2&p&q \\
{ – 3}&q&{ – p + r} \\
{12}&r&{ – q + 3r}
\end{array}} \right] = \left[ {5\,\,\,b\,\,c} \right]$ , then minimum value of $(b + c)$ is