Matrices

$\Delta =0$

sin 3 θ= $\frac{}{}$ $\frac{1}{2}$

but $0<\theta <\frac{\pi }{2}$ $\frac{}{}$

θ= 70°

$A^2=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

$=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$

$a\leftrightarrow$ R2

$=-\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$

$R_2\leftrightarrow R_3$

$=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ =1 

$\begin{array}{l}{B}_0=A^{49}+2A^{98}=A+2I\\ B_n=Adj\left(B_n-1\right)\\ B_4=Adj\left(Adj\left(Adj\left(Adj{B}_0\right)\right)\right)\end{array}$

$={\left|B_0\right|}^{{\left(n-1\right)}^4}={\left|B_0\right|}^{16}$

$B_0=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]+\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$

$=2\left(4-0\right)-1\left(0-1\right)=9$

$B_4{\left(9\right)}^{16}=3^{32}$

Kindly consider the following figure

Here,

$A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \end{array}\right]$            

we get A² = A and similarly, for

$B=A-I=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill \end{array}\right]$            

We get B² = -B $\Rightarrow$  B³ = B

$\therefore A^n+{\left(\omega B\right)}^n=A+{\left(\omega B\right)}^{n}forn\in N$

For ${\omega }^n$  to be unity n shall be multiple of 3 and for Bⁿ to be B. n shell be 3, 5, 7, ….9

$\therefore n=\left\{3,9,15,.....99\right\}$     

Number of elements = 17.

$A+B=\left[\begin{array}{cc}\hfill \beta +1\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill \alpha \hfill \end{array}\right]$

${\left(A+B\right)}^2=\left[\begin{array}{cc}\hfill {\left(\beta +1\right)}^2\hfill & \hfill 0\hfill \\ \hfill 3\left(\beta +1\right)+3\alpha \hfill & \hfill {\alpha }^2\hfill \end{array}\right]$

$\therefore \left[\begin{array}{cc}\hfill 1\hfill & \hfill -\alpha +1\hfill \\ \hfill 2\alpha +4\hfill & \hfill {\alpha }^2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill {\left(\beta +1\right)}^2\hfill & \hfill 0\hfill \\ \hfill 3\left(\alpha +\beta +1\right)\hfill & \hfill {\alpha }^2\hfill \end{array}\right]$

= 1 = 1

$B^2=\left[\begin{array}{cc}\hfill \beta \hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \beta \hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$

$\beta =0,\alpha =-1={\alpha }_2$

$\left|{\alpha }_1-{\alpha }_2\right|=\left|1-\left(-1\right)\right|=2$

$\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \\ \hfill c_1\hfill & \hfill c_2\hfill & \hfill c_3\hfill \end{array}\right]a_1,b_1,c_{1}t\left\{-1,0,1\right\}$

$\underset{9times}{\underset{?}{a_1+a_2+a_3+....}}=5$

Total = 414

$A=\left[\begin{array}{cc}\hfill 0\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{array}\right]$

$A^2=\left[\begin{array}{cc}\hfill 0\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 0\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{array}\right]$

$=\left[\begin{array}{cc}\hfill -4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -4\hfill \end{array}\right]=-4\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=-4l$

$N^2=\frac{{\left(2^{20}-1\right)}^2}{25}l$

$M{N}^2=\frac{4}{125}{\left(2^{20}-1\right)}^{3}l$

$=\left[\begin{array}{ccc}\hfill \lambda \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \lambda \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \lambda \hfill \end{array}\right],\lambda \ne k$

$A\left(\begin{array}{l}1\\ 1\\ 0\end{array}\right)=\left(\begin{array}{l}1\\ 1\\ 0\end{array}\right)$

$A\left(\begin{array}{l}1\\ 0\\ 1\end{array}\right)=\left(\begin{array}{l}-1\\ 0\\ 1\end{array}\right)$

$A\left(\begin{array}{l}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{l}1\\ 1\\ 2\end{array}\right)$

$\Rightarrow A\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$

$AB=C\Rightarrow A=C{B}^{-1}$

$\left|A-2l\right|=\left(-4\right)\left(-1\right)+3\left(-1\right)+1\left(-1\right)$

4 – 3 – 1 = 0

$\mathcal{l}\left(-1,0,1\right)+m\left(-1,1,0\right)=\left(-4,3,1\right)\Rightarrow -\mathcal{l}-m=-4$

$m=3,\mathcal{l}=1$

Apart from upper and lower triangular matrices, we have the unit triangular matrix, strictly-triangular matrix and an atomic-triangular matrix.

A few types of matrices are row and column matrices, singleton, null matrix, square, diagonal, scalar, identity, equal, triangular (both upper and lower), singular and non-singular matrices, symmetric, skew-symmetric, hermitian, and orthogonal matrices. NCERT excercise on matrices chapter covers questions related to this topic