Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Model Papers
0 votes

If \( A = \begin{bmatrix} cos\: \alpha & sin\: \alpha \\ -sin\: \alpha & cos\: \alpha \end{bmatrix} \), then show that AA' = I.

Can you answer this question?

1 Answer

0 votes
  • If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
  • If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
  • $cos\:^2 \alpha+sin^2\alpha=1$
  • An identity matrix or unit matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. An identity matrix of order 2, $I_{2}= \begin{bmatrix} 1 &0 \\ 0&1 \end{bmatrix}$
$A = \begin{bmatrix} cos\: \alpha & sin\: \alpha \\ -sin\: \alpha & cos\: \alpha \end{bmatrix}$
Transpose can be obtained by changing the rows into column.
$A' = \begin{bmatrix} cos\: \alpha & -sin\: \alpha \\ sin\: \alpha & cos\: \alpha \end{bmatrix}$
$AA'= \begin{bmatrix} cos\: \alpha & sin\: \alpha \\ -sin\: \alpha & cos\: \alpha \end{bmatrix}\begin{bmatrix} cos\: \alpha & -sin\: \alpha \\ sin\: \alpha & cos\: \alpha \end{bmatrix}$
$\Rightarrow \begin{bmatrix} cos\:^2 \alpha+sin^2\alpha & cos\alpha(-sin\alpha)+sin\: \alpha cos\alpha \\ -sin\: \alpha cos\alpha+cos\alpha sin\alpha & sin^2\alpha+cos\: ^2\alpha \end{bmatrix}$
$\Rightarrow \begin{bmatrix} 1 & -cos\alpha sin\alpha+sin \alpha cos\alpha \\ -sin \alpha cos\alpha+cos\alpha sin\alpha & 1 \end{bmatrix}$
$\Rightarrow \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$
$\Rightarrow AA'=I$
answered Apr 9, 2013 by sharmaaparna1

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App