Question

Find the value of : $\left[\begin{array}{ccc}1&2&3\\-4&3&6\\2&-7&9\end{array}\right]$. DONT SPAM! ​

Answer 1

We must know :-

$\sf{D=\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] }$

$\sf{\longrightarrow D=a_1 \left[\begin{array}{ccc}b_2&c_2\\b_3&c_3\end{array}\right] -b_1 \left[\begin{array}{ccc}a_2&c_2\\a_3&c_3\end{array}\right]+c_1 \left[\begin{array}{ccc}a_2&b_2\\a_3&b_3\end{array}\right] }$

$\sf{\longrightarrow D=a_1(b_2c_3-b_3c_2)-b_1(a_2c_3-a_3c_2)+c_1((a_2b_3-a_3b_2)}$

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Now, according to the given question :-

$\sf{\left[\begin{array}{ccc}1&2&3\\-4&3&6\\2&-7&9\end{array}\right] }$

$\sf{ =1 \left[\begin{array}{ccc}3&6\\-7&9\end{array}\right] -2\left[\begin{array}{ccc}-4&6\\2&9\end{array}\right]+3 \left[\begin{array}{ccc}-4&3\\2&-7\end{array}\right] }$

$\sf{=(27+42)-2(-36-12)+3(28-6)}$

$\sf{=69+96+66}$

$\sf{=231}$

Therefore, the answer is 231.

Answer 2

$\huge{\orange{\underline{\purple{\mathscr{Solution}}}}}$

In ΔABC and ΔBAD,

BC = BA (Common)

∠ABC = ∠BAD = 90°

AC = AD (Given)

, ΔABC ≅ ΔBAD [SAS congruency]

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Thus,

AC = BD [CPCT]

Diagonals are equal.

Now,

In ΔAOB and ΔCOD,

∠BAO = ∠DCO (Alternate interior angles)

∠AOB = ∠COD (Vertically opposite)

AB = CD (Given)

ΔAOB ≅ ΔCOD [AAS congruency]

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Thus,

AO = CO [CPCT].

Diagonal bisect each other.

Now,

In ΔAOB and ΔCOB,

OB = OB (Given)

AO = CO (diagonals are bisected)

AB = CB (Sides of the square)

ΔAOB ≅ ΔCOB [SSS congruency]

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also, ∠AOB = ∠COB

∠AOB + ∠COB = 180° (Linear pair)

Thus, ∠AOB = ∠COB = 90°