$3\mathrm{.}1(10$ points$)$

For the matrix A and vector y given below, compute the following:

Transpose of A $($please name it AT$)$

Matrix product of AT by A $($please name it ATA$)$

Determinant of ATA $($checking that det$($ATA$)0$ indicates that ATA is an invertible matrix$)$

Inverse of ATA $($please name it ATAi$)$

Product of AT by y $($please name it ATy$)$

Product of ATAi by ATy $($please name it x$)$

Print the value of x

$3\mathrm{.}2(5$ points$)$ Apply the np$.$linalg.lstsq method to find the least$-$squares solution to the linear matrix equation A$.$x $=$ y$.$ Print the value of x found with this method to confirm that it is the same as the one found by explicit calculation in $3\mathrm{.}1.$

$3\mathrm{.}3(5$ points$)$ For the matrix B of new observations given below, use the value of the coefficient vector x found in $3\mathrm{.}1$ and $3\mathrm{.}2$ to predict the value of y$^$ according to the linear regression model