Question:
Given $a_i^2 + b_i^2 + c_i^2 = 1$ and $a_i a_j + b_i b_j + c_i c_j = 0$ for $i \neq j$, and
\[
A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}
\]
Then $\det(AA^T) = $ ?
Given $a_i^2 + b_i^2 + c_i^2 = 1$ and $a_i a_j + b_i b_j + c_i c_j = 0$ for $i \neq j$, and
\[ A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \] Then $\det(AA^T) = $ ?
\[ A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \] Then $\det(AA^T) = $ ?
Show Hint
For orthogonal matrices, $AA^T = I$, so $\det(AA^T) = 1$
Updated On: May 18, 2025
- 0
- 1
- -1
- 3
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Given vectors are orthonormal (dot products zero for $i \ne j$ and each has magnitude 1), hence matrix $A$ is orthogonal.
So $AA^T = I \Rightarrow \det(AA^T) = \det(I) = 1$
So $AA^T = I \Rightarrow \det(AA^T) = \det(I) = 1$
Was this answer helpful?
0
0
Top Questions on Matrices
- Let \( A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} -5 \\ -2 \end{bmatrix}, C = [9 \ \ 7], \) which of the following is defined?
- The matrix $A = \begin{bmatrix} \sqrt{5} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$ is an:
- If \( A \) is a square matrix of order \( 3 \times 3 \), \( \det A = 3 \), then the value of \( \det(3A^{-1}) \) is:
- If \( A \) is a square matrix of order 2 such that \( \text{det} = 4 \), then \( \text{det}(4 \, \text{adj} \, A) \) is equal to:
- If $A$ and $B$ are two square matrices each of order 3 with $|A| = 3$ and $|B| = 5$, then $|2AB|$ is:
View More Questions
Questions Asked in AP EAPCET exam
- The number of positive integers less than 10000 which contain the digit 5 at least once is
- AP EAPCET - 2025
- Permutations
- If the de Broglie wavelength of an electron is \( 2 \, \text{nm} \), then its kinetic energy is nearly (Planck's constant \( = 6.6 \times 10^{-34} \, \text{J s} \) and mass of electron \( = 9 \times 10^{-31} \, \text{kg} \))
- AP EAPCET - 2025
- Dual nature of matter
- The domain and range of a real valued function \( f(x) = \cos (x-3) \) are respectively.
- AP EAPCET - 2025
- Functions
- If a force F = (3i - 2j) N acting on a body displaces it from point (1 m, 2 m) to point (2 m, 0 m), then work done by the force is
- AP EAPCET - 2025
- work, energy and power
- Two bodies A and B of masses 1.5 kg and 3 kg are moving with velocities 20 m/s and 15 m/s respectively. If the same retarding force is applied on the two bodies, then the ratio of the distances travelled by the bodies A and B before they come to rest is
- AP EAPCET - 2025
- work, energy and power
View More Questions