Search any question & find its solution
Question:
Answered & Verified by Expert
If $X_{4 \times 3}, Y_{4 \times 3}$ and $P_{2 \times 3}$ are the matrices then the order of the matrix $\left[P\left(X^T Y\right)^{-1} P^T\right]^T$ is
Options:
Solution:
1885 Upvotes
Verified Answer
The correct answer is:
$2 \times 2$
$\mathrm{P}_{2 \times 3}, \mathrm{X}_{4 \times 3}, Y_{4 \times 3}$
$\therefore \mathrm{X}^{\mathrm{T}}$ will be of order $3 \times 4$
and $\mathrm{P}^{\mathrm{T}}$ will be of order $3 \times 2$
$\therefore$ Order of $\mathrm{X}^{\mathrm{T}} \mathrm{Y}=3 \times 3$
$\therefore$ Order of $\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1}=3 \times 3$
Now, order of $\mathrm{P}\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1}=2 \times 3$
$\therefore$ Order of $\mathrm{P}\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1} \mathrm{P}^{\mathrm{T}}=2 \times 2$
$\therefore$ Order of $\left[\mathrm{P}\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1} \mathrm{P}^{\mathrm{T}}\right]^{\mathrm{T}}=2 \times 2$
$\therefore \mathrm{X}^{\mathrm{T}}$ will be of order $3 \times 4$
and $\mathrm{P}^{\mathrm{T}}$ will be of order $3 \times 2$
$\therefore$ Order of $\mathrm{X}^{\mathrm{T}} \mathrm{Y}=3 \times 3$
$\therefore$ Order of $\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1}=3 \times 3$
Now, order of $\mathrm{P}\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1}=2 \times 3$
$\therefore$ Order of $\mathrm{P}\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1} \mathrm{P}^{\mathrm{T}}=2 \times 2$
$\therefore$ Order of $\left[\mathrm{P}\left(\mathrm{X}^{\mathrm{T}} \mathrm{Y}\right)^{-1} \mathrm{P}^{\mathrm{T}}\right]^{\mathrm{T}}=2 \times 2$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.