If A = 1/3{bmatrix}1&2&2\\2&1&-2\&2&b{bmatrix} is an orthogonal matrix, then

Correct Answer is : a = – 2, b = – 1

Correct Answer is : a = – 2, b = – 1

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BITSAT 2018 Paper

Section: Mathematics

Question : 102 of 150

Marks: +1, -0

If A =

\frac{1}{3}\begin{bmatrix}1&2&2\\2&1&-2\\a&2&b\end{bmatrix}

is an orthogonal matrix, then

Solution:

As A is an orthogonal matrix,

AA^T

= I

⇒

\frac{1}{3}\begin{bmatrix}1&2&2\\2&1&-2\\a&2&b\end{bmatrix}

\frac{1}{3}\begin{bmatrix}1&2&a\\2&1&2\\2&-2&b\end{bmatrix}

\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}

⇒

\frac{1}{9}\begin{bmatrix}1&2&2\\2&1&-2\\a&2&b\end{bmatrix}

\begin{bmatrix}1&2&a\\2&1&2\\2&-2&b\end{bmatrix}

\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}

⇒

\begin{bmatrix}9&0&a+4+2b\\0&9&2a+2-2b\\a+4+2b&2a+2-2b&a^2+4+b^2\end{bmatrix}

\begin{bmatrix}9&0&0\\0&9&0\\0&0&9\end{bmatrix}

⇒ a + 4 + 2b = 0, 2a + 2 – 2b = 0 and

a^2

+ 4 +

b^2

= 9

⇒ a + 2b + 4 = 0, a – b + 1 = 0 and

a^2 + b^2

= 5

⇒ a = – 2, b = – 1

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