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If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = An Bn.
CBSE | Class 12 | Exercise 1 | Maths | Matrices | NCERT Exemplar
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = An Bn.

Let

    \[~P\left( n \right)\text{ }:\text{ }{{\left( AB \right)}^{n}}~=\text{ }{{A}^{n}}{{B}^{n}}\]

In this way,

    \[P\left( 1 \right)\text{ }:\text{ }{{\left( AB \right)}^{1}}~=\text{ }{{A}^{1}}{{B}^{1}}\]

    \[AB\text{ }=\text{ }AB\]

In this way,

    \[P\left( 1 \right)\]

is valid.

Let

    \[P\left( n \right)\]

is valid for some

    \[k\in N\]

Now,

NCERT Exemplar Solutions Class 12 Mathematics Chapter 3 - 91

Subsequently,

    \[P\left( 1 \right)\]

is valid and at whatever point

    \[P\left( k \right)\]

is valid

    \[P\left( k\text{ }+\text{ }1 \right)\]

is valid.

In this way,

    \[P\left( n \right)\]

is valid for all

    \[n\in N.\]

 

i

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