If \(A = \left[ {\begin{array}{*{20}{c}} {\sin\alpha }&{ - \cos \alpha }\\ {\cos\alpha }&{\sin \alpha } \end{array}} \right]\), then for what value of α, A is an identity matrix?

Concept: 

Diagonal Matrix:

Any square matrix in which all the elements are zero except those in the principal diagonal is called a diagonal matrix.

i.e A = [aij]n × n is a diagonal matrix if aij = 0 for i not equal to j.

Identity Matrix:

A diagonal matrix in which all the principal diagonal elements are equal to 1 is called an identity matrix. It is also known as unit matrix whereas an identity matrix of order n is denoted by I or In

Calculation:

Given: \(A = \left[ {\begin{array}{*{20}{c}} {\sin\alpha }&{ - \cos \alpha }\\ {\cos\alpha }&{\sin \alpha } \end{array}} \right]\)

Here, we have to find the value of α such that A is an identity matrix.

i.e A = I

\(⇒ \left[ {\begin{array}{*{20}{c}} {\sin α }&{ - \cos α }\\ {\cosα }&{\sin α } \end{array}} \right] = \;\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]\)

⇒ sin α = 1,

⇒ cos α = 0

⇒ α = 90°

∴ The required value is 90° .