Let a- b- c- are three unit vectors inclined with each other by an angle of -3- If a-a1-a2-a3k- - b-b1-b2-b3k- - c-c1-c 2- c 3k-and A be the matrix of order 3 - 3 such that adjadj- A - begin bmatrix a1 - a2 - a3 11 b1 - b2 - b3U c1 - c2 - c311 lend bmatrix then - A -4-A- 1-2B- 1-21C- 1-2 -2D- 1-4

The correct option is A 1/√(2)|adj(adj A)|=|A|^4 adj(adj A) = [ a_1 amp; a_2 amp; a_3; nbsp; b_1 amp; b_2 amp; b_3; nbsp; c_1 amp; c_2 amp; c_3; ...

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Byju's Answer

Standard XII

Mathematics

Definition of a Determinant

Let a⃗, b⃗, c...

Question

Let $\to a, \to b, \to c$ are three unit vectors inclined with each other by an angle of $\frac{π}{3}$ . If $\to a = a_{1}^i + a_{2}^j + a_{3}^k; \to b = b_{1}^i + b_{2}^j + b_{3}^k; \to c = c_{1}^i + c_{2}^j + c_{3}^k$
and $A$ be the matrix of order $3 \times 3$ such that
$a d j (a d j A) = ⎡ ⎢ ⎣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ⎤ ⎥ ⎦$ then $| A |^{4} =$

\frac{1}{\sqrt{2}}

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\frac{1}{2}

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\frac{1}{2 \sqrt{2}}

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\frac{1}{4}

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Solution

The correct option is A $\frac{1}{\sqrt{2}}$
$| a d j (a d j A) | = | A |^{4}$

$a d j (a d j A) = ⎡ ⎢ ⎣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ⎤ ⎥ ⎦ \Rightarrow | a d j (a d j A) |^{2} = {∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2} = [\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a . \to a & \to a . \to b & \to a . \to c \to b . \to a & \to b . \to b & \to b . \to c \to c . \to a & \to c . \to b & \to c . \to c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \frac{1}{2} & 1 & \frac{1}{2} \frac{1}{2} & \frac{1}{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = \frac{1}{2} | A |^{8} = \frac{1}{2}$

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Similar questions

Let $\to a = a_{1}^i + α_{2}^j + a_{3}^k, \to b = b_{1}^i + b_{2}^j + b_{3}^k a n d \to c = c_{1}^i + c_{2}^j + c_{3}^k$ be three non-zero vectors such that $\to c$ is a unit vector perpendicular to both the vectors $\to a$ and $\to b$ . If the angle between $\to a$ and $\to b$ is $\frac{π}{6},$
then ${∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}$ is equal to