Chapter 3

A college keeps a record in the form of a matrix (that is, a table) of the number of students receiving various grades in various courses each year. What does the sum of these matrices represent? What does their difference represent?

Solve the following sets of equations by reducing the matrix to row echelon form. \(\left\{$$\begin{array}{l}2x+y=4\\ 7x-2y=3\end{array}$$\right.\)

Find $AB$ and $BA$ given $$A=\left(\begin{array}{ll}1& 2\\ 3& 6\end{array}\right),B=\left(\begin{array}{rr}10& 4\\ -5& -2\end{array}\right)$$ Observe that $AB$ is the null matrix; if we call it 0 , then $AB=0$, but neither $A$ nor $B$ is Show that $A$ is singular.

All lines are in the $(x,y)$ plane. Find the slope of the line whose paramerric equation is $r=(i-j)+(2i+3j)t$.

Find $AB,BA,A+B,5A,3B,5A-3B$. Observe that $AB\ne BA$. Show that $\det (AB)=\det (BA)=(\det A)(\det B)$, but that $\det (A+B)\ne \det A+\det B,\det 5A\ne$ 5 det $A$, and det $3B\ne 3$ det $B$. (In Problem 2, show that det $3B=9$ det $B$, and in Problem 3 . det $3B=27\det B.)$ $$A=\left(\begin{array}{rr}2& -5\\ -1& 3\end{array}\right),B=\left(\begin{array}{rr}-1& 4\\ 0& 2\end{array}\right)$$

Given the matrix $$A=\left(\begin{array}{ccc}1& 0& 5i\\ -2i& 2& 0\\ 1& 1+i& 0\end{array}\right)$$ find the transpose, the adjoint, the inverse, the complex conjugate, and the transpose conjugate of $A$. Verify that $A{A}^{-1}=A^{-1}A=$ the unit matrix.

Use vectors to prove the following theorems from geometry; The diagonals of a parallelogram bisect each other.

Find $AB,BA,A+B,5A,3B,5A-3B$. Observe that $AB\ne BA$. Show that $\det (AB)=\det (BA)=(\det A)(\det B)$, but that $\det (A+B)\ne \det A+\det B,\det 5A\ne$ 5 det $A$, and det $3B\ne 3$ det $B$. (In Problem 2, show that det $3B=9$ det $B$, and in Problem 3 . det $3B=27\det B.)$ $$A=\left(\begin{array}{rrr}1& 0& 2\\ 3& -1& 0\\ 0& 5& 1\end{array}\right),B=\left(\begin{array}{rrr}1& 1& 0\\ 0& 2& 1\\ 3& -1& 0\end{array}\right)$$

Solve the given set of equations by reducing the matrix to echelon form. Sa! geometrically what the solution is (one point, all points on a line or on a plane, or no solution). If the solution is a line, write its vector equation. \(\left\{$$\begin{array}{r}x+y-2z=7\\ 2x+y-4z=11\\ x-y-2z=1\end{array}$$\right.\)

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \left\{$$\begin{array}{r}-x+y-z=4\\ x-y+2z=3\\ 2x-2y+4z=6\end{array}$$\right. $$