Determinants

Part 1.  Basic Properties of Determinants

Enter matrices A and B:

\(A\ \  = 
      \ {\ {2, \(-6\), 0, 6}, \ {5, \(-7\), 8, 2}, \ 

        	\ \ \ \ \ \ \ \ \ \ 	{\(-3\), \(-2\), 4, 1}, 
        \ {5, \(-1\), 4, \(-1\)}\ }; \ \ A\  // \ MatrixForm\)\(B\  = 
      \ {\ {4, 20, \(-29\), 25}, \ {3, 13, \(-24\), 13}, 

        		\ \ \ \ \ \ \ \ {1, 5, \(-7\), 4}, \ {5, 25, \(-35\), 21}\ }; \ \ 
    B\  // \ MatrixForm\)

Enter identity and zero matrices:

I4  =  IdentityMatrix[4];   I4  //  MatrixFormI9  =  IdentityMatrix[9];   I9  //  MatrixFormZ4  =  Table[0, {4}, {4}];   Z4  //  MatrixFormZ9  =  Table[0, {9}, {9}];   Z9  //  MatrixForm

Enter random 7 x 7 matrices R and S:

\(R\  = \ Table[Random[Integer, \ {\(-99\), 99}], \ {7}, {7}]; 

    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R\  // \ MatrixForm\)\(S\  = \ Table[Random[Integer, \ {\(-99\), 99}], \ {7}, {7}\ ]; 

    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S\  // \ MatrixForm\)

Enter a random 4 x 4 matrix P:

\(P\  = \ Table[Random[Integer, \ {\(-99\), 99}], {4}, {4}]; 

    P\  // \ MatrixForm\)

Enter matrix K:

\(K\  = 
      \ {\ {3, 1, \(-2\), 5}, \ {\(-2\), 6, \(-15\), 2}, 

        		\ \ \ \ \ \ \ \ {8, \(-5\), 7, 9}, \ {5, \(-6\), 9, 4}\ }; \ \ 
    K\  // \ MatrixForm\)

Part 2.  Row and Columns Operations

(You will find % often useful in this part of the lab.)

Enter matrices U and L:

\(U\  = 
      \ {\ {2, \(-9\), 23, 5}, \ {0, \(-3\), \ 4, 33}, 

        		\ \ \ \ \ \ \ \ {0, 0, 1, 71}, \ {0, 0, 0, 5}\ }; \ \ 
    U\  // \ MatrixForm\)\(L\  = 
      \ {\ {15, 0, 0, 0}, \ {27, 1, 0, 0}, 

        		\ \ \ \ \ \ \ \ {34, 9, 6, 0}, \ {99, 7, 45, \(-1\)}\ }; \ \ 
    L // MatrixForm\)

Enter a random 4 x 4 matrix T:

\(T\  = \ Table[Random[Integer, \ {\(-99\), 99}], \ {4}, {4}\ ]; 

    T\  // \ MatrixForm\)

Part 3.  The Adjoint Matrix

Part 4.  Hilbert Matrices (optional)

Enter the defintion:

H  =  HilbertMatrix[6];   H  //  MatrixForm

The commands below plot vectors corresponding to the columns of the 3 x 3 Hilbert matrix.  Rotate the 3-dimensional graph (by changing the ViewPoint) to determine whether or not the vectors are coplanar.

\(\(Show[
      Graphics3D[{\ Line[\ {{0, 0, 0}, {1, 1/2, 1/3}}\ ], \ 
          Line[\ {{0, 0, 0}, {1/2, 1/3, 1/4}}\ ], \ 
          Line[\ {{0, 0, 0}, {1/3, \ 1/4, \ 1/5}}\ ]\ }], \ 
      ViewPoint -> \ {.5, 1, .5}]
\)\)

The N command converts the fractional entries of the matrix to decimal entries.

NH  = N[H];  NH //  MatrixForm

Part 5.  Summary