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If A And B Are Square Matrices of the Same Order Such That Ab = Ba, Then Prove by Induction that Ab" = B"A. Further, Prove that (Ab)" = A"B" for All N ∈ N - Mathematics

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

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Solution

A and B are square matrices of the same order such that AB = BA.

To prove  P(n) : AB" = B"A, `n in N`

For n = 1, we have:

Therefore, the result is true for n = 1.

Let the result be true for n = k.

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have AB" = B"A, `n in N`

Now, we prove that (AB)" = A"B" for all n ∈ N

For n = 1, we have:

Therefore, the result is true for n = k + 1.

Thus, by the principle of mathematical induction, we have (AB)" = A"B", for all natural numbers.

 

Concept: Types of Matrices
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NCERT Class 12 Maths
Chapter 3 Matrices
Q 12 | Page 101
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