There is a special type of non-singular matrices which are widely used in applications of matrices to geometry. For simplicity, we consider two-dimensional analytical geometry.

**Application of matrices to Geometry**

There is a special type of non-singular matrices which are widely
used in applications of matrices to geometry. For simplicity, we consider
two-dimensional analytical geometry.

Let *O *be the origin, and *x *'*O x
*and *y *'*Oy *be the *x *-axis and *y *-axis.
Let *P *be a point in the plane whose coordinates are (*x*, *y*)
with respect to the coordinate system. Suppose that we rotate the *x *-axis and *y
*-axis about the origin, through an angle *Î¸** *as shown in the figure. Let *X *'*OX *and
*Y *'*OY *be the new *X *-axis and new *Y *-axis. Let
( *X *,*Y *) be the new set of coordinates of *P *with
respect to the new coordinate system. Referring to
Fig.1.1, we get

x = OL = ON âˆ’ LN = X cos Î¸ â€“ QT = X cos Î¸ âˆ’ Y sin Î¸ ,

y = PL = PT + TL = QN + PT =
X sin Î¸ + Y cos Î¸ .

These equations provide transformation of one coordinate system
into another coordinate system.

The above two equations can be written in the matrix form

Hence, we get the
transformation *X *= *x *cos*Î¸
*- *y *sin*Î¸ *, *Y *= *x *sin*Î¸
*+ *y *cos*Î¸ *.

This transformation is
used in **Computer
Graphics **and determined by the matrix

We note
that the matrix *W *satisfies a special
property *W *^{-1} = *W ^{T} *; that is,

A square matrix *A *is called **orthogonal **if *AA ^{T} *=

*A *is **orthogonal **if and only if *A *is non-singular and *A*^{âˆ’}^{1}* ***=**** A^{T} .**

Prove that is orthogonal.

Similarly,
we get A^{T}A = I_{2} . Hence AA^{T} = A^{T}A =
I_{2} â‡’
A is orthogonal.

If A = , is orthogonal, find *a*, *b *and *c *,
and hence *A*^{âˆ’}^{1}.

If A is orthogonal, then AA^{T} = A^{T} A = I_{3}
. So, we have

Tags : Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Application of matrices to Geometry | Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix

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