Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 5E from Chapter 1.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 5E
Chapter:
Problem:
Describe the possible echelon forms of a nonzero 2 × 2 matrix. Use the symbols ∎, * and 0, as in the first part of Example 1.
Step-by-Step Solution
Given Information
We have to write the possible echelon form of a nonzero $2 \times 2$ matrix by using the symbols $\blacksquare$, $*$ and 0
Step-1: the possibility-1
The $2 \times 2$ matrix can have 2 pivot columns, the possible form is: \[M = \left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&{\blacksquare} \end{array}} \right]\]
Step-2: the possibility-2
The $2 \times 2$ matrix can have 1 pivot columns, the possible forms are: \[\begin{array}{l} M = \left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&0 \end{array}} \right]\\ M = \left[ {\begin{array}{*{20}{c}} 0&{\blacksquare}\\ 0&0 \end{array}} \right] \end{array}\]
Step-3: The three possible forms are
\[M = \left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&{\blacksquare} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&0 \end{array}} \right],\,\,\left[ {\begin{array}{*{20}{c}} 0&{\blacksquare}\\ 0&0 \end{array}} \right]\]
We have to write the possible echelon form of a nonzero $2 \times 2$ matrix by using the symbols $\blacksquare$, $*$ and 0
Step-1: the possibility-1
The $2 \times 2$ matrix can have 2 pivot columns, the possible form is: \[M = \left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&{\blacksquare} \end{array}} \right]\]
Step-2: the possibility-2
The $2 \times 2$ matrix can have 1 pivot columns, the possible forms are: \[\begin{array}{l} M = \left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&0 \end{array}} \right]\\ M = \left[ {\begin{array}{*{20}{c}} 0&{\blacksquare}\\ 0&0 \end{array}} \right] \end{array}\]
Step-3: The three possible forms are
\[M = \left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&{\blacksquare} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} {\blacksquare}&*\\ 0&0 \end{array}} \right],\,\,\left[ {\begin{array}{*{20}{c}} 0&{\blacksquare}\\ 0&0 \end{array}} \right]\]