Q. 63.7( 27 Votes )

# Find the values of x, y, z if the matrix satisfy the equation A’A =I.

Answer :

Given A

Transpose of a matrix: If A be an m×n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. It is denoted by A’ or AT.

Transpose of A = A’ =

Given equation

A’A =I

=

=

. =

As these matrices are equal to each other that means each element of matrix on L.H.S is equal each element of matrix on R.H.S.

On comparing elements on both sides we get

4y2 + z2 = 1 …… (1)

2y2 – z2 = 0 …… (2)

x2 + y2 + z2 = 1 …… (3)

– y2 – z2 = 0 …… (4)

From equation (4) we get,

x2 = y2 + z2 …… (5)

Substituting this value in equation (3) we get,

2y2 + 2z2 = 1 …… (6)

Subtracting equation (2) and (6) we get,

3z2 = 1

z2 = 1/3

z = 1/3

Substituting value of z in equation (2) we get,

2y2 = 1/3

y2 = 1/6

y = �1/6

Substituting values of y and z in equation (5) we get,

x2 = 1/2

x = 1/2

Hence values of x, y, z are 1/2, 1/6, 1/3 respectively.

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