Matrices and vectors

We are given the matrix #\:A \:# and the special column-vectors #\:\vec{u} \:# and #\:\vec{v}\:# below. #\:# Answer the following matrix multiplication questions.

\[ \:A = \matrix{-7 & -13 \\ 14 & -1 \\ } \:\:\:\:\:\:\:\:\:\:\: \vec{u} = \matrix{0 \\ 7 \\ } \:\:\:\:\:\:\:\:\: \vec{v} = \matrix{4 \\ 0 \\ } \]

Use the matrix button #\:# #\:# to enter each answer. This exercise is programmed only to use the matrix button. Sorry for the extra step, but it will not get marked correctly if you use the vector button: #\:#

Step 1.	#\:A\vec{u}\:=\:\matrix{-7 & -13 \\ 14 & -1 \\ } \matrix{0 \\ 7 \\ }\:=#
Step 2.	#\:A\vec{v}\:=\:\matrix{-7 & -13 \\ 14 & -1 \\ } \matrix{4 \\ 0 \\ }\:=#
Step 3.	Observation. #\:# Suppose that #\:A\:# is a matrix with two columns and suppose \[\:\vec{r} = \left( \begin{array}{c} a \\ b \end{array} \right)\:\] Then the following is true. #\:# When we calculate # A-\vec{r} # # \vec{r}-A # # A+\vec{r} # # \vec{r}\hspace{0.05 cm}A # # A\hspace{0.05 cm}\vec{r} # we will obtain a multiple of the #\mbox{ second }# column of the matrix #\:A,\:# if we make: # a=0 \: \: \mbox{ and }\: \: b=0 # # a\neq0 \: \: \mbox{ and }\: \: b\neq0 # # a\neq0 \: \: \mbox{ and }\: \: b=0 # # a=0 \: \: \mbox{ and }\: \: b\neq0 #

Edna's Math Class for OpenStax

Matrices and vectors: Intro to Matrix Multiplication