3.5.1 Lab 6¶
Part I. One-to-one Transformation¶
The given matrix is the standard matrix of a linear transformation. Determine if the linear transformation is one-to-one.
If your birthday is in:
January through April, do matrix \(J\).
May through August, do matrix \(M\).
September through December, do matrix \(S\).
\[\begin{split}J = \left[\begin{array}{rrr}4&7&6\\-2&-2&-1\\4&4&1\\\end{array}\right]\end{split}\]
J = [4 7 6 ; -2 -2 -1 ; 4 4 1 ] ;
\[\begin{split}M = \left[\begin{array}{rrr}4&7&6\\-2&-2&-1\\4&4&1\\\end{array}\right]\end{split}\]
M = [4 7 6 ; -2 -2 -1 ; 4 4 1 ] ;
\[\begin{split}S = \left[\begin{array}{rrr}4&7&6\\-2&-2&-1\\4&4&1\\\end{array}\right]\end{split}\]
S = [4 7 6 ; -2 -2 -1 ; 4 4 1 ] ;
Part II. Matrix Multiplication¶
Using MATLAB’s dot function and not the multiplication operation, calculate the products of the following matrices.
If your first name begins with:
A through D, do \(A * B\).
D through J, do \(D * E\).
J through Z, do \(J * K\).
\[\begin{split}A = \left[\begin{array}{rrrr}0&8&1&5\\3&1&8&-2\\\end{array}\right]
\hspace{1cm}\text{and}\hspace{1cm} B = \left[\begin{array}{rrr}-1&-2&-2\\7&4&0\\0&-2&5\\0&-2&3\\\end{array}\right]\end{split}\]
A = [0 8 1 5 ; 3 1 8 -2 ];
B = [-1 -2 -2 ; 7 4 0 ; 0 -2 5 ; 0 -2 3 ];
\[\begin{split} D = \left[\begin{array}{rrrr}-2&1&-1&-2\\7&3&-2&6\\\end{array}\right]\hspace{1cm}\text{and}\hspace{1cm} E = \left[\begin{array}{rrr}-1&-2&-2\\7&4&0\\0&-2&5\\0&-2&3\\\end{array}\right]\end{split}\]
D = [-2 1 -1 -2 ; 7 3 -2 6 ];
E = [-1 -2 -2 ; 7 4 0 ; 0 -2 5 ; 0 -2 3 ];
\[\begin{split}J = \left[\begin{array}{rrrr}-2&-1&1&-2\\-1&-2&7&0\\\end{array}\right] \hspace{1cm}\text{and}\hspace{1cm} K = \left[\begin{array}{rr}2&0\\3&-1\\3&6\\8&2\\\end{array}\right] \end{split}\]
J = [-2 -1 1 -2 ; -1 -2 7 0 ];
K = [2 0 ; 3 -1 ; 3 6 ; 8 2 ];
Part III: Inverse Matrix¶
Use row reduction (showing all steps) to find the inverse matrix.
If your last name begins with:
B through G, do B.
H through Q, do H.
R through Z, do R.
A pick any of the three.
\[\begin{split}B = \left[\begin{array}{rrr}6&-8&-5\\6&-7&-3\\3&-4&-2\\\end{array}\right]\end{split}\]
B = [6 -8 -5;6 -7 -3;3 -4 -2] ;
\[\begin{split}H = \left[\begin{array}{rrr}-3&-3&13\\-1&-2&3\\1&2&-5\\\end{array}\right]\end{split}\]
H = [-3 -3 13 ; -1 -2 3;1 2 -5] ;
\[\begin{split}R = \left[\begin{array}{rrr}1&0&-8\\-2&0&19\\-2&-2&11\\\end{array}\right]\end{split}\]
R = [1 0 -8;-2 0 19;-2 -2 11];