91Ó°ÊÓ

1\. Find the signed area of the parallelogram formed by the following pairs of vectors in \(\mathbb{R}^{2}\). "a. \(\mathbf{x}=(1,5), \mathbf{y}=(2,3)\) b. \(\mathbf{x}=(4,3), \mathbf{y}=(5,4)\) c. \(\mathbf{x}=(2,5), \mathbf{y}=(3,7)\)

Find the signed volume of the parallelepiped formed by the following triples of vectors in \(\mathbb{R}^{3}\). "a. \(\mathbf{x}=(1,2,1), \mathbf{y}=(2,3,1), \mathbf{z}=(-1,0,3)\) b. \(\mathbf{x}=(1,1,1), \mathbf{y}=(2,3,4), \mathbf{z}=(1,1,5)\) c. \(\mathbf{x}=(3,-1,2), \mathbf{y}=(1,0,-3), \mathbf{z}=(-2,1,-1)\)

Without using Proposition 1.7, show that for any elementary matrix \(E\), we have \(\operatorname{det} E^{\mathrm{T}}=\operatorname{det} E\). (Hint: Consider each of the three types of elementary matrices.)

Show that if the entries of a matrix \(A\) are integers, then det \(A\) is an integer. (Hint: Use induction.)

a. Suppose \(A\) is an \(n \times n\) matrix with integer entries and det \(A=\pm 1\). Show that \(A^{-1}\) has all integer entries. b. Conversely, suppose \(A\) and \(A^{-1}\) are both matrices with integer entries. Prove that \(\operatorname{det} A=\pm 1\).

a. If \(C\) is the cofactor matrix of \(A\), give a formula for \(\operatorname{det} C\) in terms of \(\operatorname{det} A\). b. Let \(C=\left[\begin{array}{rrr}1 & -1 & 2 \\ 0 & 3 & 1 \\ -1 & 0 & -1\end{array}\right]\). Can there be a matrix \(A\) with cofactor matrix \(C\) and det \(A=3\) ? Find a matrix \(A\) with positive determinant and cofactor matrix \(C\).