If A = left( {begin{array}{*{20}{c}}1&a0&1end{array}} right) , then A^n (where n ∈ N ) equ

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

normal

If $A =$ $\left( {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right)$ , then $A^n$ (where $n \in N$) equals

$\left( {\begin{array}{*{20}{c}}1&{na}\\0&1\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}1&{{n^2}a}\\0&1\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}1&{na}\\0&0\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}n&{na}\\0&n\end{array}} \right)$

Solution

We have $A^2 =$ $\left( {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right)$ $\left( {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right)$ $=$ $\left({\begin{array}{*{20}{c}}1&{2a}\\0&1\end{array}} \right)$

$A^3$ $= A^2A$ $=$ $\left( {\begin{array}{*{20}{c}}1&{2a}\\0&1\end{array}}\right)$ $\left( {\begin{array}{*{20}{c}}1&a\\0&1\end{array}} \right)$ $=$ $\left({\begin{array}{*{20}{c}}1&{3a}\\0&1\end{array}} \right)$

In general by induction, $A^n =$ $\left( {\begin{array}{*{20}{c}}1&{na}\\0&1\end{array}} \right)$ , $\forall \, n \in N$

Standard 12

Mathematics

Suppose $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is a real matrix with non-zero entries, $\alpha d-b c=0$ and $A^2=A$. Then, $a + d$ equals

hard

(KVPY-2018)

Find the value of $a, \,b,\, c,$ and $d$ from the equation: $\left[\begin{array}{cc}a-b & 2 a+c \\ 2 a-b & 3 c+d\end{array}\right]=\left[\begin{array}{cc}-1 & 5 \\ 0 & 13\end{array}\right]$

easy

If $A=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]$ and $B=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right],$ then $A B=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]$. and $\mathrm{BA}=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right] .$ Clearly $\mathrm{AB} \neq \mathrm{BA}$.

Thus matrix multiplication is not commutative.

medium

Given $3\left[\begin{array}{ll}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right],$ find the values of $x, \,y, \,z$ and $w$.

medium

Let $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in R$ and $A^{4}=\left[a_{i j}\right] .$ If $a_{11}=109,$ then $a_{22}$ is equal to