Let A be a 2 × 2 real matrix with entries from {0,1} and |mathrm{A}| ≠ 0 . Consider following two

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3 and 4 .Determinants and Matrices

hard

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0,1\}$ and $|\mathrm{A}| \neq 0 .$ Consider the following two statements :

$(P)$ If $A \neq I_{2},$ then $|A|=-1$

$(\mathrm{Q})$ If $|\mathrm{A}|=1,$ then $\operatorname{tr}(\mathrm{A})=2$

where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$ Then

$(P)$ is true and $(\mathrm{Q})$ is false

Both $(P)$ and $(Q)$ are false

Both $(P)$ and $(Q)$ are true

$(P)$ is false and $(Q)$ is true

(JEE MAIN-2020)

Solution

$|A| \neq 0$

For $(\mathrm{P}): \mathrm{A} \neq \mathrm{I}_{2}$

So, $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ or $\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]$ or $\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right]$ or $\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$

or $\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$

IAl can be -1 or 1

So (P) is false.

For $(\mathrm{Q}) ; \quad|\mathrm{A}|=1$

$A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ or $\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ or $\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$

$\Rightarrow \operatorname{tr}(\mathrm{A})=2$

$\Rightarrow \mathrm{Q}$ is true

Standard 12

Mathematics

Let $A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$. If $M$ and $N$ are two matrices given by $M =\sum \limits_{ k =1}^{10} A ^{2 k }$ and $N =\sum \limits_{ k =1}^{10} A ^{2 k -1}$ then $MN ^{2}$ is

medium

(JEE MAIN-2022)

Let $x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ and $A =\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$. For $k \in N$, if $X ^{\prime} A ^{ k } X =33$, then $k$ is equal to.

normal

(JEE MAIN-2022)

Let $A$ be a square matrix such that $A A^T=I$. Then $\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to

medium

(JEE MAIN-2024)

If $A = \left( {\begin{array}{{20}{c}}
{\alpha – 1}\\
0\\
0
\end{array}} \right),\,\,\,B = \left( {\begin{array}{{20}{c}}
{\alpha + 1}\\
0\\
0
\end{array}} \right)$ be two matrices, then $AB^T$ is a non-zero matrix for $\left| \alpha \right|$ not equal to

hard

(AIEEE-2012)

If $ A$ is a square matrix, then $A + {A^T}$ is

easy

Solution

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Let $A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$. If $M$ and $N$ are two matrices given by $M =\sum \limits_{ k =1}^{10} A ^{2 k }$ and $N =\sum \limits_{ k =1}^{10} A ^{2 k -1}$ then $MN ^{2}$ is

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If $A = \left( {\begin{array}{{20}{c}}
{\alpha – 1}\\
0\\
0
\end{array}} \right),\,\,\,B = \left( {\begin{array}{{20}{c}}
{\alpha + 1}\\
0\\
0
\end{array}} \right)$ be two matrices, then $AB^T$ is a non-zero matrix for $\left| \alpha \right|$ not equal to