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  • Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 62 maths textbook solution

Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 62 maths textbook solution

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Answer: Hence proved, A^{n}=\operatorname{diag}\left(a^{n} \quad b^{n} \quad c^{n}\right) for all positive integer n.

Hint: We use the principle of mathematical induction.

Given:A=\operatorname{diag}\left(\begin{array}{lll}a & b & c\end{array}\right)

Prove: A^{n}=\operatorname{diag}\left(a^{n} \quad b^{n} \quad c^{n}\right) for all positive integer n             …(i)

Solution: step 1:

 put n=1 in eqn (i)

A^{1}=\operatorname{diag}\left(a^{1} \quad b^{1} \quad c^{1}\right) \ \ [ x^1 = x ] 

A=\operatorname{diag}\left(\begin{array}{lll}a & b & c\end{array}\right)

So, A^n is true for n=1.

Step 2: let A^n be true for n=k, so

A^{k}=\operatorname{diag}\left(a^{k} \quad b^{k} \quad c^{k}\right)

Step 3: now, we have to show that

A^{k+1}=\operatorname{diag}\left(a^{k+1} \quad b^{k+1} \quad c^{k+1}\right)

 

Now,

\\ \begin{array}{l} A^{k+1}=A^{k} A \\\\ =d i a g\left(a^{k} \quad b^{k} \quad c^{k}\right) d i a g(a \quad b \quad c) \\\\ A^{k+1}=\left[\begin{array}{ccc} a^{k} & 0 & 0 \\ 0 & b^{k} & 0 \\ 0 & 0 & c^{k} \end{array}\right]\left[\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right] \\\\\\ =\left[\begin{array}{ccc} a^{k} \times a+0+0 & 0+0+0 & 0+0+0 \\ 0+0+0 & 0+b^{k} \times b+0 & 0+0+0 \\ 0+0+0 & 0+0+0 & 0+0+c^{k} \times c \end{array}\right] \\\\ =\left[\begin{array}{ccc} a^{k+1} & 0 & 0 \\ 0 & b^{k+1} & 0 \\ 0 & 0 & c^{k+1} \end{array}\right] \\\\\\ A^{k+1}=\operatorname{diag}\left(a^{k+1} \quad b^{k+1} \quad c^{k+1}\right) \end{array}

So, A^n is true for n=k+1 whenever A^n is true for n=k

Hence, by principle of mathematical induction A^n is true for all positive integers.

Posted by

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