Let A = left[ {begin{array}{*{20}{c}}1&2&22&1&22&2&1end{array}}right]

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3 and 4 .Determinants and Matrices

normal

Let $A =$ $ \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&2\\2&2&1\end{array}}\right]$ , then

$A^2 - 4A - 5I_3 = 0$

$A^{-1} = \frac{1}{5} (A - 4I_3)$

$A^2$ is invertible

All of the above

Solution

$A^2 =$ $ \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&2\\2&2&1\end{array}}\right]$ $ \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&2\\2&2&1\end{array}}\right]$ $=$ $\left[ {\begin{array}{*{20}{c}}9&8&8\\8&9&8\\8&8&9\end{array}} \right]$
We have $A^2 – 4A – 5I_3$

$=$ $\left[ {\begin{array}{*{20}{c}}9&8&8\\8&9&8\\8&8&9\end{array}} \right]$ $- 4$ $ \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&2\\2&2&1\end{array}}\right]$ $- 5$ $\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]$ $= O$

==> $5I_3 = A^2 – 4A = A(A – 4I_3)$

==>$I_3 = A$ $\left[ {\frac{1}{5}(A – 4{I_3})} \right]$

==> $A^{-1} =$ $\frac{1}{5}(A – 4{I_3})$

Note that $|A| = 5$. Since $|A^3| = |A|^3 = 5^3 \ne 0, A^3$ is invertibleSimilarly, $A^2$ is invertible

Standard 12

Mathematics

Find the values of $x, y, z$ if the matrix $A=\left[\begin{array}{ccc}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]$ satisfy the equation $A^{\prime} A=I$.

hard

For the matrix $A=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right],$ verify that $(A-A^{\prime})$ is a skew symmetric matrix

easy

Let $A$ is a symmetric and $ \,B$ is a skew symmetric matrix, such that $A – B = \left[ {\begin{array}{*{20}{c}}
1&2 \\
3&4
\end{array}} \right]$, then $|A|$ is

normal

Let $A$ be a $2 \times 2$ matrix with real entries such that $A ^{\prime}=\alpha A + I$, where $\alpha \in R -\{-1,1\}$. If det $\left(A^2-A\right)=4$, then the sum of all possible values of $\alpha$ is equal to

hard

(JEE MAIN-2023)

Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname{det}(Q)=9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:

medium

(JEE MAIN-2021)