Let, #[A]_(2xx4)=[(2,3,5,7),(13,17,19,23)]#,
#[B]_(4xx3)=[(64,28,-18),(-64,-27,18),(15,5,-5),(0,0,1)]#,
#[C]_(3xx1)=[(1),(u),(v)]#.
Then, #[A]_(2xx4)*[B]_(4xx3)*[C]_(3xx1)# is defined, and, is a
#(2xx1)# Matrix. .
Now, #[B]_(4xx3)*[C]_(3xx1)# is a #4xx1# Matrix, given by,,
#=[(64+28u-18v),(-64-27u+18v),(15+5u-5v),(v)]#.
Next, #[A]_(2xx4)*{[B]_(4xx3)*[C]_(3xx1)}# is a #2xx1# Matrix
and, #[A]_(2xx4)*{[B]_(4xx3)*[C]_(3xx1)}#,
#=[(2,3,5,7),(13,17,19,23)]*[(64+28u-18v),(-64-27u+18v),(15+5u-5v),(v)]#.
Now,
#2(64+28u-18v)+3(-64-27u+18v)+5(15+5u-5v)+7(v),#
#=(128-192+75)+u(56-81+25)+v(-36+54-25+7)#,
#=11#, and,
#13(64+28u-18v)+17(-64-27u+18v)+19(15+5u-5v)+23(v),#
#=(832-1088+285)+u(364-459+95)+v(-234+306-95+23)#,
#=29#.
#:.[(2,3,5,7),(13,17,19,23)]*[(64+28u-18v),(-64-27u+18v),(15+5u-5v),(v)]=[(11),(29)]#.
From here on, consider the variables exchange
#alpha = 3 + u - v#
#beta = u#
#ABC = K#
#ADE = K#
#D = ((10, 18, 10), (-10, -18, -9), (0, 5, 0), (3, -1, 1)), E = ((1), (alpha), (beta))#
#DE = BC#
#E = MC#
#DMC = BC => DM = B => M = ((1, 0, 0), (3, 1, -1), (0, 1, 0))#
