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ANSC*6370 QG and Animal Models
Fall 03 - Quiz 4 - Answers

1.

Given that

(a)

What value of x would yield a zero determinant for ${\bf A}$?

x = -2.

(b)

Using the value of x from above, what is the rank of ${\bf A}$?

Rank is 2.

(c)

Derive a generalized inverse of ${\bf A}$ using the value of x from above.

There are an infinite number of possible generalized inverses. One special one is derived from inverting the upper 2 by 2 submatrix.

$${\bf A}^{-} = \frac{-1}{8} \left( \begin{array}{rrr} 1 & -3 & 0 \\ -2 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right).$$

2.

Show how you would create a matrix with 5 rows and 5 columns that

(a)

Has rank of 0?

Matrix of all zeros.

(b)

Has rank of 1?

Matrix of all ones (or any constant, -1, 2, 3617, etc.).

(c)

Has rank of 2?

Create a five by two matrix of rank two, then pre-multiply it times its transpose (or times another matrix of order 2 by 5 with rank 2).

Or create any two by two matrix with rank 2, say ${\bf D}$, then let

$${\bf C} = \left( \begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 1 & -1 \\ 3 & -4 \\ -63.1 & 72.2998 \end{array} \right),$$

and calculate ${\bf CDC}'$.

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