Question

Let $$ A=\left[\begin{array}{rr} 1 & 3 \\ -2 & -1 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 3 & -4 \\ 2 & -2 \end{array}\right] $$ a. Find $A^{T}$ and show that $\left(A^{T}\right)^{T}=A$. b. Show that $(A+B)^{T}=A^{T}+B^{T}$. c. Show that $(A B)^{T}=B^{T} A^{T}$.

          Let
$$
A=\left[\begin{array}{rr}
1 & 3 \\
-2 & -1
\end{array}\right] \text { and } B=\left[\begin{array}{ll}
3 & -4 \\
2 & -2
\end{array}\right]
$$
a. Find $A^{T}$ and show that $\left(A^{T}\right)^{T}=A$.
b. Show that $(A+B)^{T}=A^{T}+B^{T}$.
c. Show that $(A B)^{T}=B^{T} A^{T}$.

Added by Jennifer J.

Instant Answer

Solved by Expert Shahina -

Step 1

$A^{T}=\left[\begin{array}{rr} 1 & -2 \\ 3 & -1 \end{array}\right]$. To show that $\left(A^{T}\right)^{T}=A$, we simply need to show that the transpose of the transpose of $A$ is equal to $A$. That is, $\left(\left(A^{T}\right)^{T}\right)_{ij}=\left(A\right)_{ij}$ Show more…

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Let $$ A=\left[\begin{array}{rr} 1 & 3 \\ -2 & -1 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 3 & -4 \\ 2 & -2 \end{array}\right] $$ a. Find $A^{T}$ and show that $\left(A^{T}\right)^{T}=A$. b. Show that $(A+B)^{T}=A^{T}+B^{T}$. c. Show that $(A B)^{T}=B^{T} A^{T}$.