If A is an idempotent matrix, then (I + A)^4 is (where I is identity matrix of order same as A )

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3 and 4 .Determinants and Matrices

hard

If $A$ is an idempotent matrix, then $(I + A)^4$ is (where $I$ is identity matrix of order same as $A$ )

$I + 11A$

$I + 8A$

$I + 17A$

$I + 15A$

Solution

Since $A^{2}=A \Rightarrow A^{3}=A \quad \Rightarrow \quad A^{4}=A$

$\therefore \quad(\mathrm{I}+\mathrm{A})^{4}$

$\quad = {\,^4}{{\rm{C}}_0}{{\rm{I}}^4} + {\,^4}{{\rm{C}}_1}{\rm{A}} + {\,^4}{{\rm{C}}_2}{{\rm{A}}^2} + {\,^4}{{\rm{C}}_3}{{\rm{A}}^3} + {\,^4}{{\rm{C}}_4}{{\rm{A}}^4}$

$=\mathrm{I}+15 \mathrm{A}$

Standard 12

Mathematics

Let $A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right)$. Then the sum of the diagonal elements of the matrix $( A + I )^{11}$ is equal to:

hard

(JEE MAIN-2023)

$AB = 0$, if and only if

normal

Find $\mathrm{AB},$ if $\mathrm{A}=\left[\begin{array}{rr}0 & -1 \\ 0 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}3 & 5 \\ 0 & 0\end{array}\right]$.

easy

Find $X$ and $Y$, if

$X+Y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$ and $X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

medium

$A$ is a $2 × 2$ matrix such that $A$ $\left[ {\begin{array}{{20}{c}}1\\{ – 1} \end{array}} \right]$ $=$$\left[ {\begin{array}{{20}{c}}{ – 1}\\2\end{array}} \right]$ and $A^2$ $\left[ {\begin{array}{{20}{c}}1\\{ – 1}\end{array}} \right]$ $=$ $\left[ {\begin{array}{{20}{c}}1\\0\end{array}} \right]$ . The sum of the elements of $A$, is