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

Chapter: 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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.

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 θ 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 − LN = X cos θ – QT =  X cos θ − Y sin θ ,

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


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 cosθ - y sinθ , Y = x sinθ + y cosθ .

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

We note that the matrix W satisfies a special property W -1 = WT ; that is, WW T = WTW = I .

 

Definition 1.3

A square matrix A is called orthogonal if AAT = ATA = I.

 

Note

A is orthogonal if and only if A is non-singular and A1 = AT .

 

Example 1.11

Prove that  is orthogonal.

Solution


Similarly, we get ATA = I2 . Hence AAT = ATA = I2 A is orthogonal.

 

Example 1.12

If A = , is orthogonal, find a, b and c , and hence A1.

Solution

If A is orthogonal, then AAT = AT A = I3 . 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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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants


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