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  • Explain solution for rd sharma class 12 chapter Algebra of matrices exercise 4.3 question 61 math

Explain solution for rd sharma class 12 chapter Algebra of matrices exercise 4.3 question 61 math

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Answer: Hence proved, A^{n+1}=B^{n}(B+(n+1) C for every integer n \in N.

Hint: We use the principle of mathematical induction.

Given: B, C are n rowed square matrix,

\\A=B+C, B C=C B, C^{2}=0, \\\\

A=B+C

Squaring both sides, we get

\\A^{2}=(B+C)^{2}\\\\ A^{2}=(B+C)(B+C)\\\\ A^{2}=B \times B+B C+C B+C           [using distributive property]

A^{2}=B^{2}+B C+B C+C^{2}        [using BC=CB] given and put value of C^{2}

\\A^{2}=B^{2}+2 B C+0\\\\ A^{2}=B^{2}+2 B C \quad \ldots(i)\\\\ A^{2}=B(B+2 C)

Now consider,P(n)=A^{n+\mathbf{1}}=B^{n}[B+(n+1) C]

Step 1:  to prove P(1) is true, put n=1

\\A^{\mathbf{1 + 1}}=B^{\mathbf{1}}[B+(1+1) C]\\\\ A^{2}=B[B+2 C]\\\\ A^{2}=B^{2}+2 B C

From equation i, P(1)is true

Step 2:  suppose P(k) is true

A^{k+1}=B^{k}[B+(k+1) C] ... (ii)

Step 3 : now we need to show that P(k+1) is true

That is we need to prove that

A^{k+2}=B^{k+1}[B+(k+2) C]

Now,

\begin{aligned} A^{k+2} &=A^{k} A^{2} \\ &=B^{(k-1)}[B+k C] \times[B(B+2 C)] \\ &=B^{k}[B+k C] \times[B+2 C] \\ &=B^{k}\left[B \times B+B \times 2 C+k C \times B+2 k C^{2}\right] \\ &=B^{k}\left[B^{2}+2 B C+k B C+2 k \times 0\right] \quad\left[\text { since } B C=C B, C^{2}=0\right] \\ &=B^{k}\left[B^{2}+B C(2+k)\right] \\ &=B^{k} \times B[B+(k+2) C] \\ &=B^{k+1}[B+(k+2) C] \end{aligned}      

So, P(n) is true for n=k+1 whenever P(n) is true for n=k.

Therefore, by principle of mathematical induction P(n) is true for all natural number.

Posted by

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