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Question: If $A = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$, then $A^4 =$...

If A=[1010]A = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, then A4=A^4 =

A

[1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

B

[1100]\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}

C

[0011]\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}

D

[0110]\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

Answer

None of the provided options is correct.

Explanation

Solution

We are given:

A=[1010]A=\begin{bmatrix}1&0\\1&0\end{bmatrix}.

Step 1. Compute A2A^2:

A2=A⋅A=[1⋅1+0⋅11⋅0+0⋅01⋅1+0⋅11⋅0+0⋅0]=[1010]=AA^2 = A\cdot A = \begin{bmatrix}1\cdot1+0\cdot1 & 1\cdot0+0\cdot0\\ 1\cdot1+0\cdot1 & 1\cdot0+0\cdot0\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 1 & 0\end{bmatrix} = A.

Since A2=AA^2 = A, by induction it follows that An=Afor any n≥1A^n=A\quad \text{for any }n\ge1.

Thus, A4=[1010]A^4 = \begin{bmatrix}1&0\\1&0\end{bmatrix}.

None of the given options is correct.