The total number of matrices $A=\left[\begin{array}{ccc}0 & 2 x & 2 x \\ 2 y & y & -y \\ 1 & -1 & 1\end{array}\right]$; $(x, y \in R, x \neq y)$ for which $A^{\top} A=3 I_{3}$

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Matrices

The total number of matrices

A = 0 2 y 1 2 x y - 1 2 x - y 1

;

(x, y \in R, x \neq = y)

for which

A^{⊤} A = 3 I_{3}

Answer: (D)

The condition

A^{T} A = 3 I_{3}

says the columns of

A

are mutually perpendicular and each has squared length

3

. Carrying out the product

A^{T} A

, the diagonal entries come out as

8 x^{2}

6 y^{2}

and

3

, while the off-diagonal entries vanish on their own. Matching the diagonal to

3

gives

8 x^{2} = 3

and

6 y^{2} = 3

, so

x = \pm \frac{3}{8}

and

y = \pm \frac{1}{2}

. Each sign is independent and the requirement

x \neq = y

excludes none of these, so there are

2 \times 2 = 4

such matrices.

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Single correct option