1. Tardigrade
  2. Question
  3. Mathematics
  4. If A = beginpmatrix2&-1 -7&4 endpmatrix and B = beginpmatrix4 1 7 2 endpmatrix then which statement is true?

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $A = \begin{pmatrix}2&-1\\ -7&4\end{pmatrix}$ and $B =\begin{pmatrix}4 & 1 \\ 7 & 2\end{pmatrix}$ then which statement is true?

Matrices

Solution:

Here $A A ^{ T }= \begin{pmatrix}2 & -1 \\ -7 & 4\end{pmatrix} \begin{pmatrix}2 & -7 \\ -1 & 4\end{pmatrix} \neq \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$
$\left( BB ^{ T }\right)_{11}=(4)^{2}+(1)^{2} \neq 1$
$( AB )_{11}=8-7=1,( BA )_{11}=8-7=1$
$\therefore A \vec{B} \neq$ BA may be not true.
Now, $AB = \begin{pmatrix}2 & -1 \\ -7 & 4\end{pmatrix} \begin{pmatrix}4 & 1 \\ 7 & 2\end{pmatrix}$
$= \begin{pmatrix}8-7 & 2-2 \\ -28+28 & -7+8\end{pmatrix}= \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} ;( AB )^{ T }= \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$