10.11 Sample test : matrices i

(9 points)

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]-2\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ . What are $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ , $h$ , and $i$ ?

(9 points)

Matrix $\left[A\right]$ is $\left[\begin{array}{ccc}4& 2& 0\\ -6& 8& \text{10}\end{array}\right]$ . What is $A+A+\frac{1}{2}A$ ?

(9 points)

Using the same Matrix $\left[A\right]$ , what is $2\frac{1}{2}A$ ?

(9 points)

$\left[\begin{array}{cc}4& 6\\ -5& 3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ \text{22}\end{array}\right]$ . What are $x$ and $y$ ?

(9 points)

$\left[\begin{array}{cccc}1& 3& 4& n\end{array}\right]\left[\begin{array}{c}2\\ 5\\ 6\\ 7\end{array}\right]$ =

(9 points)

$\left[\begin{array}{cc}4& -2\\ 0& 3\\ n& 1\end{array}\right]\left[\begin{array}{ccc}3& 6& 8\\ 2& 9& 0\\ 1& -1& 4\\ 0& 4& 2\end{array}\right]$ =

(9 points)

$\left[\begin{array}{ccc}3& 6& 8\\ 2& 9& 0\\ 1& -1& 4\\ 0& 4& 2\end{array}\right]\left[\begin{array}{cc}4& -2\\ 0& 3\\ n& 1\end{array}\right]$ =

(9 points)

$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{c}\text{some}\\ \text{matrix}\end{array}\right]=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$ . What is "some matrix"?

(5 points)

$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{c}\text{some}\\ \text{matrix}\end{array}\right]=\left[\begin{array}{ccc}c& b& a\\ f& e& d\\ i& h& g\end{array}\right]$ . What is "some matrix"?

(8 points)

• A

  Write two matrices that can be added and can be multiplied.

• B

  Write two matrices that cannot be added or multiplied .

• C

  Write two matrices that can be added but cannot be multiplied .

• D

  Write two matrices that can be multiplied but cannot be added .

(15 points)

• A

  Find the inverse of the matrix $\left[\begin{array}{cc}4& x\\ 1& -2\end{array}\right]$ by using the definition of an inverse matrix.
  If you are absolutely flat stuck on part (a), ask for the answer. You will receive no credit for part (a) but you may then be able to go on to parts (b) and (c).

• B

  Test it, by showing that it fulfills the definition of an inverse matrix.

• C

  Find the inverse of the matrix $\left[\begin{array}{cc}4& 3\\ 1& -2\end{array}\right]$ by plugging $x=3$ into your answer to part (a).

Extra credit: