The total number of matrices A = beginpmatrix0&2y&1 2x&y&-1 2x&-y&1 endpmatrix ,(x,y ∈ R, x ≠ y) for which ATA = 3I3 is :

Question

The total number of matrices A = beginpmatrix0&2y&1 2x&y&-1 2x&-y&1 endpmatrix ,(x,y ∈ R, x ≠ y) for which ATA = 3I3 is : (A) 6 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

ATA =3I3 beginpmatrix0&2x&2x 2y&y&-y 1&-1&1 endpmatrix beginpmatrix0&2y&1 2x&y&-1 2x&-y&1 endpmatrix = beginpmatrix3&0&0 0&3&0 0&0&3 endpmatrix beginpmatrix8x2&0&0 0&6y2&0 0&0&3 endpmatrix = [3&0&0 0&3&0 0&0&3] 8x2= 3 6y2=3 x2 =3/8 y2 =1/2 x = ± √(3/8) ; y= ± √(1/2)

Q. The total number of matrices A=⎝⎛​02x2x​2yy−y​1−11​⎠⎞​,(x,y∈R,x=y) for which ATA=3I3​ is :

6

2

3

4

ATA=3I3​ ⎝⎛​02y1​2xy−1​2x−y1​⎠⎞​⎝⎛​02x2x​2yy−y​1−11​⎠⎞​=⎝⎛​300​030​003​⎠⎞​ ⎝⎛​8x200​06y20​003​⎠⎞​=⎣⎡​300​030​003​⎦⎤​ 8x2=3 6y2=3 x2=3/8 y2=1/2 x=±83​​;y=±21​​

Q. The total number of matrices $A = ⎝ ⎛ 0 2 x 2 x 2 y y - y 1 - 1 1 ⎠ ⎞, (x, y \in R, x \neq = y)$ for which $A^{T} A = 3 I_{3}$ is :