Chapter 23 Multiple choice questions

* not completed

. Write an effective Hamiltonian matrix using Hückel theory for:



The α and β coefficients for nitrogen are 1.5 and 1.75, respectively.
(Use the numbering system given.)

${\displaystyle \left(\begin{array}{ccccccc}\alpha & \beta & 0& 0& 0& 0& 0\\ \beta & \alpha & \beta & 0& 0& 0& 0\\ 0& \beta & \alpha & \beta & 0& 0& 1.75\beta \\ 0& 0& \beta & \alpha & \beta & 0& 0\\ 0& 0& 0& \beta & \alpha & \beta & 0\\ 0& 0& 0& 0& \beta & \alpha & 1.75\beta \\ 0& 0& 1.75\beta & 0& 0& 1.75\beta & 1.5\alpha \end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccccccc}\alpha & \beta & 0& 0& 0& 0& 0\\ \beta & \alpha & \beta & 0& 0& 0& 0\\ 0& \beta & \alpha & \beta & 0& 0& \beta \\ 0& 0& \beta & \alpha & \beta & 0& 0\\ 0& 0& 0& \beta & \alpha & \beta & 0\\ 0& 0& 0& 0& \beta & \alpha & \beta \\ 0& 0& \beta & 0& 0& \beta & 1.5\alpha \end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccccccc}\alpha & \beta & 0& 0& 0& 0& \beta \\ \beta & \alpha & \beta & 0& 0& 0& 0\\ 0& \beta & \alpha & 0& 0& 0& 1.75\beta \\ 0& 0& \beta & \alpha & \beta & 0& 0\\ 0& 0& 0& \beta & \alpha & \beta & 0\\ 0& 0& 0& 0& \beta & \alpha & 1.75\beta \\ 0& 0& 1.75\beta & 0& 0& 1.75\beta & 1.5\alpha \end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccccccc}\alpha & \beta & 0& 0& 0& 0& 0\\ \alpha & \beta & \alpha & 0& 0& 0& 0\\ 0& \alpha & \beta & \alpha & 0& 0& 1.75\alpha \\ 0& 0& \alpha & \beta & \alpha & 0& 0\\ 0& 0& 0& \alpha & \beta & \alpha & 0\\ 0& 0& 0& 0& \alpha & \beta & 1.75\alpha \\ 0& 0& 1.75\alpha & 0& 0& 1.75\alpha & 1.5\beta \end{array}\right)}$ correct incorrect

* not completed

. If $\mathbf{A}={\displaystyle \left(\begin{array}{ccc}1& 4& 1\\ 2& 7& 3\\ 4& 2& 5\end{array}\right)}$ and $\mathbf{B}={\displaystyle \left(\begin{array}{ccc}2& 0& 9\\ 4& 3& 9\\ 5& 7& 8\end{array}\right)}$, then calculate: B - A.

${\displaystyle \left(\begin{array}{ccc}3& 4& 10\\ 6& 10& 12\\ 9& 9& 13\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}1& -4& 8\\ 2& -4& 6\\ 1& 5& 3\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}-1& 4& -8\\ -2& 4& -6\\ -1& -5& -3\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}1& 4& 8\\ 2& 4& 6\\ 1& 5& 3\end{array}\right)}$ correct incorrect

* not completed

. Determine the following matrix scalar product:
$3a{\displaystyle \left(\begin{array}{ccc}2b& 1& 7c\\ 3& 6b& 4a\end{array}\right)}$.

${\displaystyle \left(\begin{array}{ccc}6ab& 3a& 21ac\\ 9a& 18ab& 12a^2\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}6ab& 1& 7c\\ 9a& 6b& 4a\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}6ab& 3a& 21ac\\ 3& 6b& 4a\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}6ab& 1& 7c\\ 3& 6b& 4a\end{array}\right)}$ correct incorrect

* not completed

. If $\mathbf{A}={\displaystyle \left(\begin{array}{ccc}-2& 5& 1\\ 0& 3& 2\end{array}\right)}$ and $\mathbf{B}={\displaystyle \left(\begin{array}{cc}1& 4\\ 3\mathrm{b}& 2a\\ 2& b\end{array}\right)}$, then calculate: AB.

${\displaystyle \left(\begin{array}{cc}15b+2& -8+10a+b\\ 5+9b& 4+6a+2b\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{cc}5b& 10a+b\\ 9b-4& 2+6a+2b\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{cc}-2& 15b\\ 0& 9b\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{cc}15b& -8+10a+b\\ 4+9b& 6a+2b\end{array}\right)}$ correct incorrect

* not completed

. What is the product of matrix A and matrix B? Show that matrix A is the identity matrix.
A = ${\displaystyle \left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}$ B = ${\displaystyle \left(\begin{array}{ccc}2& -1& 3\\ 3& 1& 4\\ 4& 0& 1\end{array}\right)}$

${\displaystyle \left(\begin{array}{ccc}2& 3& 4\\ -1& 1& 0\\ 3& 4& 1\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}2& -1& 3\\ 3& 1& 4\\ 4& 0& 1\end{array}\right)}$ correct incorrect

* not completed

. Find the determinant of ${\displaystyle \left(\begin{array}{ccc}2& 5& 8\\ 5& 3& 7\\ 4& 6& 9\end{array}\right)}$ .

29 correct incorrect

199 correct incorrect

259 correct incorrect

-89 correct incorrect

* not completed

. Find the eigenvalues of the following matrix:
$\mathrm{A}={\displaystyle \left(\begin{array}{cc}-3& 2\\ 4& -5\end{array}\right)}$.

-15 and 8 correct incorrect

-3 and -5 correct incorrect

-1 and -7 correct incorrect

-23 and 23 correct incorrect

* not completed

. Write down the transpose of the following matrix:
${\displaystyle \left(\begin{array}{ccc}2& 0& 9\\ 1& 9& 7\\ 3& 3& 6\end{array}\right)}$.

${\displaystyle \left(\begin{array}{ccc}0& 2& 9\\ 9& 1& 7\\ 3& 3& 6\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}3& 3& 6\\ 1& 9& 7\\ 2& 0& 9\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}9& 7& 6\\ 0& 9& 3\\ 2& 1& 3\end{array}\right)}$ correct incorrect

${\displaystyle \left(\begin{array}{ccc}2& 1& 3\\ 0& 9& 3\\ 9& 7& 6\end{array}\right)}$ correct incorrect

* not completed

. Write down the inverse of the following matrix:
$\mathbf{P}={\displaystyle \left(\begin{array}{cc}-2& -3\\ 7& 6\end{array}\right)}$.

${\mathbf{P}}^{\mathbf{-}\mathbf{1}}\mathbf{ }={\displaystyle \left(\begin{array}{cc}\frac{-2}{9}& \frac{-1}{3}\\ \frac{7}{9}& \frac{2}{3}\end{array}\right)}$ correct incorrect

${\mathbf{P}}^{\mathbf{-}\mathbf{1}}\mathbf{ }={\displaystyle \left(\begin{array}{cc}\frac{2}{3}& \frac{1}{3}\\ \frac{-7}{9}& \frac{-2}{9}\end{array}\right)}$ correct incorrect

${\mathbf{P}}^{\mathbf{-}\mathbf{1}}\mathbf{ }={\displaystyle \left(\begin{array}{cc}\frac{2}{9}& \frac{1}{3}\\ \frac{-7}{9}& \frac{-2}{3}\end{array}\right)}$ correct incorrect

${\mathbf{P}}^{\mathbf{-}\mathbf{1}}\mathbf{ }={\displaystyle \left(\begin{array}{cc}\frac{-2}{3}& \frac{-1}{3}\\ \frac{7}{9}& \frac{2}{9}\end{array}\right)}$ correct incorrect

* not completed

. Use matrices to solve the following linear simultaneous equations:
x + 2z = 9
4x + 2y + z = 14
x + 3y + 4z = 26.

x = 3, y = -0.5, and z = 3 correct incorrect

x = -1, y = 1, and z = 6 correct incorrect

x = 1, y = 3, and z = 4 correct incorrect

x = 1, y = 4, and z = 2 correct incorrect