Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 56 maths

Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 56 maths

Answers (1)

Answer: Hence proved, A^{n}=\left[\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right] for all possible integers n.

Hint: Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

We use the principle of mathematical induction to prove. A^{n}=\left[\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right]

Given:

A^{n}=\left[\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right]

Prove:

A^{n}=\left[\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right] for all possible integers n                         …(i)

Solution:

Step 1: put n=1 in eqn (i)

A^{1}=\left[\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right]

So, A^n is true for n=1

Step 2 :let,A^n  be true for n=k, then

A^{k}=\left[\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right] ... (ii)

Step 3 : we have to show that A^{k+1}=\left[\begin{array}{cc}1 & k+1 \\ 0 & 1\end{array}\right]

So,

\\A^{k+1}=A^{k} \times A=\left[\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] \\\\\\\ A^{k+1}=\left[\begin{array}{cc}1+0 & 1+k \\ 0+0 & 0+1\end{array}\right]=\left[\begin{array}{cc}1 & 1+k \\ 0 & 1\end{array}\right]

This shows that A^n is true for n=k+1 whenever it is true for n=k

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

Posted by

infoexpert22

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Class 12 board exams 2025 to have 50% competency-based questions; no change in 10th
anam-khan

CBSE Class 12 board exams 2024 over; expected result date, trends of last 10 years
anam-khan

CBSE Class 12 accountancy board exam analysis 2024 out; paper rated easy, balanced

Latest Question