prove by induction. Please simple and detailed solution. Show each step. Consider the Fibonacci numbers defined as follows: The first two Fibonacci numbers are 0 and 1,and each subsequent Fibonacci number is the sum of the two previous ones. The n-th Fibonaccinumber is denoted Fn. In other words, the Fibonacci numbers are defined recursively by therules:Fo := 0,F₁ = 1,F₁ = Fi-1 + Fi-2, for i ≥ 2Fibonacci numbers come up naturally in several ways and are important in many applicationsbecause they have many, partially surprising properties such as the ones expressed in the fol-lowing theorems. E.g., the Fibonacci recurrence function is very useful for computing powersof matrices efficiently.Prove the following two properties about the Fibonacci numbers by induction:Given matrix multiplication defined as follows:a11 a12 (b11 b12)421 022 b21 b22,:) (₂1prove the following property by induction:Vn> 1:=n=abı +a12b21 a11b12+a12b22)a21b11 + a22b21 a21b12+ a22b22,(Fn+1FnFnFn-1)

To show that for all where is the Fibonacci number.Concepts Used:To prove that a statement…

Prove the following two properties about the Fibonacci numbers by induction: Given matrix multiplication defined as follows: (b11 b12) (anbu+a12b21 a11b12+a12b22) 021 022 b21 b22, a21b11a22b21 a21b12+ a22b22, ) (b₂ prove the following property by induction: Vn 21: a11 a12 = n Fn+1 (1 3)" = (²+¹ - Fn Fn Fn-1/