Please show all work.

Will upvote if correct

$1.$ a$.$ Convert the following into proper Hamming Code using the matrix method. First convert $5$ into binary, then put it in Hamming Code format using the matrix multiplication method. $(20$ pts$.)$

b$.$ Use the Inverse Matrix method to solve the following system of linear equations. $(20$ pts$.)$

$3$X $+$ Z $=31$

$2$X $2$Y $+$ Z $=7$

Y $+3$Z $=-9$