Question

If A is a 3×3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T = ______________.

Answer 1

Given:
A is a 3×3 non-singular matrix
AA^T = A^TA
B = A⁻¹A^T


$$\begin{gathered} \mathrm{Now}, \\ B{B}^{\mathrm{T}}=\left(A^{-1}{A}^{\mathrm{T}}\right){\left(A^{-1}{A}^{\mathrm{T}}\right)}^{\mathrm{T}} \\ =\left(A^{-1}{A}^{\mathrm{T}}\right){\left(A^{\mathrm{T}}\right)}^{\mathrm{T}}{\left(A^{-1}\right)}^{\mathrm{T}} \\ =\left(A^{-1}{A}^{\mathrm{T}}\right)\left(A\right){\left(A^{\mathrm{T}}\right)}^{-1} \\ =A^{-1}\left(A^{\mathrm{T}}A\right){\left(A^{\mathrm{T}}\right)}^{-1} \\ =A^{-1}\left(A{A}^{\mathrm{T}}\right){\left(A^{\mathrm{T}}\right)}^{-1}\left(\because A^{\mathrm{T}}A=A{A}^{\mathrm{T}}\right) \\ =\left(A^{-1}A\right)\left(A^{\mathrm{T}}{\left(A^{\mathrm{T}}\right)}^{-1}\right) \\ =\left(I\right)\left(I\right) \\ =I \end{gathered}$$


Hence, BB^T = I.