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Q60E

Expert-verified
Found in: Page 277

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Which of the following functions F of ${\mathbf{A}}{\mathbf{=}}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ are linear in both columns? Which are linear in both rows? Which are alternating on the columns? role="math" localid="1659502796300" ${\mathbit{a}}{\mathbf{.}}{\mathbf{}}{\mathbit{F}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}{\mathbf{=}}{\mathbit{b}}{\mathbit{c}}\phantom{\rule{0ex}{0ex}}{\mathbit{b}}{\mathbf{.}}{\mathbf{}}{\mathbit{F}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}{\mathbf{=}}{\mathbit{c}}{\mathbit{d}}\phantom{\rule{0ex}{0ex}}{\mathbit{c}}{\mathbf{.}}{\mathbf{}}{\mathbit{F}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}{\mathbf{=}}{\mathbit{a}}{\mathbit{c}}\phantom{\rule{0ex}{0ex}}{\mathbit{d}}{\mathbf{.}}{\mathbf{}}{\mathbit{F}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}{\mathbf{=}}{\mathbit{b}}{\mathbit{c}}{\mathbf{-}}{\mathbit{a}}{\mathbit{d}}\phantom{\rule{0ex}{0ex}}{\mathbit{e}}{\mathbf{.}}{\mathbf{}}{\mathbit{F}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}{\mathbf{=}}{\mathbit{c}}$

a) Function is linear in both columns and both rows, but it's not alternating on the columns.

b) Function is linear in both columns, but not in both rows, nor is it alternating on the columns.

c) Function is linear in both rows, but not in both columns, nor is it alternating on the columns.

d) Function is linear in both rows and both columns, as well as alternating on the columns.

e) Function is not linear in both rows nor both columns, nor is it alternating on the columns.

See the step by step solution

## Step 1: Given

Given Matrix,

$\mathrm{A}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$.

## Step 2: (a) To determine F(A)  = bc

F(A) = bc

Function F is linear in both columns and both rows, but it's not alternating on the columns.

## Step 3: (b) To determine F(A) = cd

F(A) = cd

Function is linear in both columns, but not in both rows, nor is it alternating on the columns.

## Step 4: (c) To determine F(A) = ac

$\mathrm{F}\left(\mathrm{A}\right)=\mathrm{ac}$

Function is linear in both rows, but not in both columns, nor is it alternating on the columns.

## Step 5: (d) To determine F(A) = bc - ad

$\mathrm{F}\left(\mathrm{A}\right)=\mathrm{bc}-\mathrm{ad}=\mathrm{det}\mathrm{A}$

Function F is linear in both rows and both columns, as well as alternating on the columns.

## Step 6: (e) To determine

$\mathrm{F}\left(\mathrm{A}\right)=\mathrm{c}$

Function F is not linear in both rows nor both columns, nor is it alternating on the columns.