1
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

For $$i=1,2,3$$ and $$j=1,23$$ If $$a_i^2+b_i^2+c_i^2=1, a_i a_j+b_i b_j+c_i c_j=0, \forall i \neq j$$ and $$A=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$$, then $$\operatorname{det}\left(A A^T\right)=$$

A
0
B
1
C
$$-$$1
D
3
2
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

If $$A=\frac{1}{7}\left[\begin{array}{ccc}3 & -2 & 6 \\ -6 & -3 & 2 \\ -2 & 6 & 3\end{array}\right]$$, then

A
$$A^{-1}=A$$
B
$$A^{-1}=A^T$$
C
$$A^{-1}$$ does not exist
D
$$A^{-1}=-A$$
3
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

If $$A=\left[\begin{array}{cc}\alpha^2 & 5 \\ 5 & -\alpha\end{array}\right]$$ and $$\operatorname{det}\left(A^{10}\right)=1024$$, then $$\alpha=$$

A
$$-$$2
B
$$-$$1
C
$$-$$3
D
0
4
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

Let $$A=\left[\begin{array}{ccc}5 & \sin ^2 \theta & \cos ^2 \theta \\ -\sin ^2 \theta & -5 & 1 \\ \cos ^2 \theta & 1 & 5\end{array}\right]$$. Then, maximum value of $$\operatorname{det}(A)$$ is

A
$$-125$$
B
200
C
$$-\frac{255}{2}$$
D
$$145$$
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