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10-637 Animal Models
Lesson 1 Assignment
L. R. Schaeffer, March 1999

1.
Add the following matrices together:
(a)
\( \left( \begin{array}{rr}
46 & 13 \\ 2 & 73 \\ -52 & 11 \end{array} \right) +
\left( \begin{array}{rr} 3 & 8 \\ 2 & -1 \\ 6 & -4
\end{array} \right) = \)
(b)
\( 44.8 \left( \begin{array}{rr} 3 & 8 \\ 1 & 9
\end{array} \right) -31.7 \left( \begin{array}{rr}
1 & -2 \\ 0 & 1 \end{array} \right) = \)
(c)
\( \left( \begin{array}{rrr} 3 & 6 & 5 \\ 4 & 7 & 2 \\
4 & 1 & 4 \end{array} \...
...gin{array}{rrr}
1 & -2 & -1 \\ 0 & -3 & 2 \\ 3 & 6 & 3 \end{array} \right) = \)


2.
Let \( {\bf A} = \left( \begin{array}{rrr} 1 & 0 & -1 \\
2 & -2 & 0 \\ 2 & -1 & -1 \end{array} \right) \) and \( {\bf B} = \left( \begin{array}{rrrr} 11 & 20 & 33 & 42 \\
90 & -81 & -71 & 61 \\ -51 & 41 & -32 & -23 \end{array} \right) \), then calculate the following (if possible):
(a)
\( {\bf AB} \)
(b)
\( {\bf A'B} \)
(c)
\( ( {\bf A}+{\bf A'}){\bf B} \)
(d)
\( {\bf BB'} \)
(e)
\( {\bf B'B} \)
(f)
\( tr({\bf BB'}) \)
(g)
\( tr({\bf B'B}) \)
(h)
\( {\bf B'A} \)


3.
Calculate the determinants of the following matrices:
(a)
\( \left( \begin{array}{rrr}
1 & 8 & 6 \\ 5 & 4 & -2 \\ -3 & -7 & -9 \end{array} \right) \)
(b)
\( \left( \begin{array}{rrr} 3 & 15 & 14 \\
-13 & 6 & -4 \\ 13 & -6 & 8 \end{array} \right) \)
(c)
\( \left( \begin{array}{rrrr} 2 & -1 & 1 & -1 \\
1 & 2 & -1 & 1 \\ -1 & 1 & 2 & -1 \\ 1 & -1 & 1 & 2
\end{array} \right) \)


4.
Calculate the inverses of the following matrices:
(a)
\( \left( \begin{array}{rr} 3 & -6 \\ 7 & 14.8
\end{array} \right) \)
(b)
\( \left( \begin{array}{rrr} 10 & 5 & -2 \\ 5 & 8 & 4 \\
-2 & 4 & 17 \end{array} \right) \)
(c)
\( \left( \begin{array}{rrrr} 1 & 2 & 1 & 0 \\
2 & 3 & 0 & 2 \\ 1 & 0 & 4 & 0 \\ 0 & 2 & 0 & 1
\end{array} \right) \)
(d)
Calculate the Cholesky decomposition of the following matrix, and determine the inverse of the triangular matrix and hence the inverse of the original matrix.

\begin{displaymath}\left( \begin{array}{rrrr} 4 & -2 & 8 & -10 \\
-2 & 10 & -1...
... & -10 & 21 & -27 \\ -10 & 14 & -27 & 36
\end{array} \right). \end{displaymath}

(e)
Determine the eigenvalues and eigenvectors of the previous matrix. You may use a computer routine for this question.


5.
Determine the rank and a generalized inverse of the following matrices:
(a)
\( \left( \begin{array}{rrrr}
1 & 2 & 4 & 0 \\ -2 & -3 & -1 & 1 \\ 0 & 1 & 7 & 1 \\
-2 & -2 & 6 & 2 \end{array} \right) \)
(b)
\( \left( \begin{array}{rrrr}
1 & 2 & 1 & 2 \\ 1 & 3 & 2 & 1 \\ 0 & 1 & 1 & 1 \\
-1 & 2 & 3 & 1 \end{array} \right) \)
Generate another one of the infinite number of possible generalized inverses using the formula given in the notes.



Larry Schaeffer
1999-02-26