Question

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" $$\begin{gathered} \mathit{a}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{b}\mathit{c} \\ \mathit{b}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{d} \\ \mathit{c}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{a}\mathit{c} \\ \mathit{d}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{b}\mathit{c}\mathbf{-}\mathit{a}\mathit{d} \\ \mathit{e}\mathbf{.}\mathbf{}\mathit{F}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathit{c} \end{gathered}$$

Short answer

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

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

c) Functionis 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.

Step-by-step solution

Given

Given Matrix,

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

(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.

(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.

(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.

(d) To determine F(A) = bc - ad

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

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

(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.