Thread: matrice help needed.

(1) Let A and B be non singular matrices of the same order, A^-1 =C and B^-1 = D. Write down the inverse of the matrix C^tB, in terms of A and D.

(2) An integer unimodular matrix has all entries integers, and determinant one.
(i) The identity matrix I2 is integer unimodular. Explain why, and find two more.

(ii) Is the inverse matrix of an integer unimodular matrix necessarily also an integer unimodular matrix? Is its transpose?

(iii) Is a product of two integer unimodular matrices also integer inumodular?

(iv) A homogeneous linear transformation has matrix A (integer unimodular).
* What can be said about the image of a point with integer co- ordinates?
* Find the image of the unit square (vertices (0,0), (1,0). (0,1) and (1,1,))

Originally Posted by Craige

(1) Let A and B be non singular matrices of the same order, A^-1 =C and B^-1 = D. Write down the inverse of the matrix C^tB, in terms of A and D.

notes:
$\displaystyle (AB)^{-1} = B^{-1}A^{-1}$
$\displaystyle (C^2)^{-1} = (C^{-1})^2$.. so by induction, $\displaystyle (C^t)^{-1} = (C^{-1})^t$..

now, can you do your problem?

Originally Posted by Craige

(2) An integer unimodular matrix has all entries integers, and determinant one.
(i) The identity matrix I2 is integer unimodular. Explain why, and find two more.

(ii) Is the inverse matrix of an integer unimodular matrix necessarily also an integer unimodular matrix? Is its transpose?

(iii) Is a product of two integer unimodular matrices also integer inumodular?

i. What is $\displaystyle \det I_2$?
other matrices which are unimodular:
$\displaystyle \left[ \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right]$
$\displaystyle \left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right]$
$\displaystyle \left[ \begin{array}{cc} 0 & -1 \\ 1 & 1 \end{array} \right]$
$\displaystyle \left[ \begin{array}{cc} 0 & 1 \\ -1 & 1 \end{array} \right]$
$\displaystyle \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$

ii. Let $\displaystyle A$ be a unimodular matrix.
Then, $\displaystyle 1=\det I = \det (AA^{-1}) = \det A \det A^{-1} $... (continue)

iii. Let $\displaystyle A,B$ be unimodular matrices.
Then $\displaystyle \det (AB) = \det A \det B$..(continue)