If left[ {begin{array}{*{20}{c}}1&2&33&1&22&3&1end{array}} right],left[ begi

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3 and 4 .Determinants and Matrices

easy

If $\left[ {\begin{array}{{20}{c}}1&2&3\\3&1&2\\2&3&1\end{array}} \right]\,\left[ \begin{array}{l}x\\y\\z\end{array} \right] = \left[ {\begin{array}{{20}{c}}4&{ - 2}\\0&{ - 6}\\{ - 1}&2\end{array}} \right]\,\left[ \begin{array}{l}2\\1\end{array} \right]$, then $(x,y,z)$=

$( - 4,\,2,\,2)$

$(4,\, - 2,\, - 2)$

$(4,\,2,\,2)$

$( - 4,\, - 2,\, - 2)$

Solution

(a) $\left[ {\begin{array}{*{20}{c}}1&2&3\\3&1&2\\2&3&1\end{array}} \right]\,\,\left[ \begin{array}{l}x\\y\\z\end{array} \right] = \left[ {\begin{array}{*{20}{c}}4\\0\\{ – 1}\end{array}\,\,\begin{array}{*{20}{c}}{ – 2}\\{ – 6}\\2\end{array}} \right]\,\left[ \begin{array}{l}2\\1\end{array} \right]\,$

$\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,x + 2y + 3z = 6\\\, \Rightarrow \,\,\,\,\,3x + y + 2z = – \,6\\\,{\rm{ }}2x + 3y + z = 0\end{array}$

On Simplification the above equation, we get the required result i.e.,

$x = – 4,\,y = 2,\,z = 2$.

Standard 12

Mathematics

If $A$ and $B$ are square matrices of order $2$, then ${(A + B)^2} = $

normal

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)

If $A$ is nilpotent matrix of index $2$, then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)

normal

The solution $(s)$ of the equation $\left| {\begin{array}{*{20}{c}}x&a&b\\ a&x&a\\b&b&x\end{array}} \right|$ $= 0$ is/are :

normal

If $\omega \ne 1$ is cube root of unity and $H = \left[ {\begin{array}{*{20}{c}}\omega &0\\0&\omega \end{array}} \right]$ then ${H^{70}}$ is equal to

medium

(AIEEE-2011)

If $\left[ {\begin{array}{*{20}{c}}1&2&3\\3&1&2\\2&3&1\end{array}} \right]\,\left[ \begin{array}{l}x\\y\\z\end{array} \right] = \left[ {\begin{array}{*{20}{c}}4&{ - 2}\\0&{ - 6}\\{ - 1}&2\end{array}} \right]\,\left[ \begin{array}{l}2\\1\end{array} \right]$, then $(x,y,z)$=

Solution

Similar Questions

If $A$ and $B$ are square matrices of order $2$, then ${(A + B)^2} = $

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

If $A$ is nilpotent matrix of index $2$, then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)

The solution $(s)$ of the equation $\left| {\begin{array}{*{20}{c}}x&a&b\\ a&x&a\\b&b&x\end{array}} \right|$ $= 0$ is/are :

If $\omega \ne 1$ is cube root of unity and $H = \left[ {\begin{array}{*{20}{c}}\omega &0\\0&\omega \end{array}} \right]$ then ${H^{70}}$ is equal to

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If $\left[ {\begin{array}{{20}{c}}1&2&3\\3&1&2\\2&3&1\end{array}} \right]\,\left[ \begin{array}{l}x\\y\\z\end{array} \right] = \left[ {\begin{array}{{20}{c}}4&{ - 2}\\0&{ - 6}\\{ - 1}&2\end{array}} \right]\,\left[ \begin{array}{l}2\\1\end{array} \right]$, then $(x,y,z)$=