5.59 Some Maple Matrix operations

Create a new matrix, by appending some rows of one matrix to rows from another matrix:

restart; with(LinearAlgebra): 
A:=< <1|2|3> , <4|5|6> >;
 

\[ \left [ \begin {array}{ccc} 1&2&3\\ \noalign {\medskip }4&5&6 \end {array} \right ] \]

B:=< <7|8|10> , <11|12|13> , <14|15|16>  >;
 

\[ \left [ \begin {array}{ccc} 7&8&10\\ \noalign {\medskip }11&12&13 \\ \noalign {\medskip }14&15&16\end {array} \right ] \]

Now append first row of A to last 2 rows of B

C:=<  A[1,1..-1] ,  B[2..-1,1..-1] >;
 

\[ \left [ \begin {array}{ccc} 1&2&3\\ \noalign {\medskip }11&12&13 \\ \noalign {\medskip }14&15&16\end {array} \right ] \]

# Now append first column of A to first 2 rows  of B 
A[1..-1,1]; 
B[1..2,1..-1]; 
C:=< A[1..-1,1] | B[1..2,1..-1]  >;
 

\[ \left [ \begin {array}{cccc} 1&7&8&10\\ \noalign {\medskip }4&11&12&13 \end {array} \right ] \]

#Now remove the middle row of B 
B; 
B:=<B[1,1..-1] , B[-1,1..-1] >;
 

\[ \left [ \begin {array}{ccc} 7&8&10\\ \noalign {\medskip }14&15&16 \end {array} \right ] \]

#now set the diagonal elements of B to be 0 
B:=RandomMatrix(3); 
for i from 1 to 3 do 
    B[i,i]:=0; 
end do: 
B;
 

\[ B:=\left [ \begin {array}{ccc} 0&99&92\\ \noalign {\medskip }8&0&-31 \\ \noalign {\medskip }69&44&0\end {array} \right ] \]

\[ \left [ \begin {array}{ccc} 0&99&92\\ \noalign {\medskip }8&0&-31 \\ \noalign {\medskip }69&44&0\end {array} \right ] \]

To find inverse.

restart; 
with(LinearAlgebra): 
A:=Matrix( [ [2,0],[4,2] ]); 
MatrixInverse(A);
 

\[ \left [ \begin {array}{cc} 1/2&0\\ \noalign {\medskip }-1&1/2 \end {array} \right ] \]

To check that for any matrix A, then A*transpose(A) is always a matrix which is symmetrical

A:=RandomMatrix(2,3); 
A.Transpose(A);
 

\[ A:=\left [ \begin {array}{ccc} 99&44&-31\\ \noalign {\medskip }29&92&67 \end {array} \right ] \]

\[ \left [ \begin {array}{ccc} 99&44&-31\\ \noalign {\medskip }29&92&67 \end {array} \right ] \]

how to create a random lower triangular matrix?

restart; 
with(LinearAlgebra); 
A:=RandomMatrix(4,4,outputoptions=[shape=triangular[lower]]);
 

\[ \left [ \begin {array}{cccc} 67&0&0&0\\ \noalign {\medskip }-31&92&0&0 \\ \noalign {\medskip }44&29&99&0\\ \noalign {\medskip }69&8&27&-4 \end {array} \right ] \]