Question

The notation Ak meas the matrix A Multiplied with itself k times (a) For the 2×2 identity matrix I, show that I2 =I (b)For the n×n identity matrix I, show that I2 =I (c) what do you think the enteries of Ik are?

Answer 1

Answer:

Entries of I^k are are also identity elements.

Step-by-step explanation:

a) For the 2×2 identity matrix I, show that I² =I

$I^{2}=\left[\begin{array}{cc}1&0\\0&1\end{array}\right] \times \left[\begin{array}{cc}1&0\\0&1\end{array}\right] \\\\=\left[\begin{array}{cc}1\times 1+0\times 0&1\times 0+0\times 1\\0\times 1+1\times 0&0\times 0+1\times1\end{array}\right] \\\\=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

Hence proved  I² =I

b) For the n×n identity matrix I, show that I² =I

n×n identity matrix is as shown in figure

Elements of identity matrix are

$\delta I_{ij}=1\quad if\quad i=j\\\delta I_{ij}=0\quad if\quad i\ne j$

As square of 1 is equal to 1 so for n×n identity matrix I, I² =I

(c) what do you think the enteries of Ik are?

As mentioned above

$\delta I_{ij}=1\quad if\quad i=j\\\delta I_{ij}=0\quad if\quad i\ne j$

Any power of 1 is equal to 1 so kth power of 1 is also 1. According to this Ik=I