$A=\left[\begin{array}{cc}7& 4\\ 7& 7\end{array}\right]$

The inverse, ${A}^{-1}$ is ?

Damien Horton
2022-07-31
Answered

Find the inverse of the following matrix A, if possible. Check that $A{A}^{-1}=I$ and ${A}^{-1}A=I$

$A=\left[\begin{array}{cc}7& 4\\ 7& 7\end{array}\right]$

The inverse, ${A}^{-1}$ is ?

$A=\left[\begin{array}{cc}7& 4\\ 7& 7\end{array}\right]$

The inverse, ${A}^{-1}$ is ?

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Izabelle Frost

Answered 2022-08-01
Author has **13** answers

$\frac{1}{ab-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

$\frac{1}{49-28}\left[\begin{array}{cc}7& -4\\ -7& 7\end{array}\right]$

Answer:

$\left[\begin{array}{cc}1/3& -4/21\\ -1/3& 1/3\end{array}\right]$

$\frac{1}{49-28}\left[\begin{array}{cc}7& -4\\ -7& 7\end{array}\right]$

Answer:

$\left[\begin{array}{cc}1/3& -4/21\\ -1/3& 1/3\end{array}\right]$

Elisabeth Esparza

Answered 2022-08-02
Author has **1** answers

$A=\left[\begin{array}{cc}7& 4\\ 7& 7\end{array}\right],\text{}find\text{}{A}^{-1}$.

Inverse of a $2\times 2$ matrix:

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{}then\text{}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$. If ad-bc=0, then A has no inverse.

So, ${A}^{-1}=\frac{1}{7\ast 7-4\ast 7}\left[\begin{array}{cc}7& -4\\ -7& 7\end{array}\right]=\frac{1}{21}\left[\begin{array}{cc}7& -4\\ -7& 7\end{array}\right]=\left[\begin{array}{cc}\frac{7}{21}& -\frac{4}{21}\\ -\frac{7}{21}& \frac{7}{21}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{3}& -\frac{4}{21}\\ -\frac{1}{3}& \frac{1}{3}\end{array}\right]$

Inverse of a $2\times 2$ matrix:

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{}then\text{}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$. If ad-bc=0, then A has no inverse.

So, ${A}^{-1}=\frac{1}{7\ast 7-4\ast 7}\left[\begin{array}{cc}7& -4\\ -7& 7\end{array}\right]=\frac{1}{21}\left[\begin{array}{cc}7& -4\\ -7& 7\end{array}\right]=\left[\begin{array}{cc}\frac{7}{21}& -\frac{4}{21}\\ -\frac{7}{21}& \frac{7}{21}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{3}& -\frac{4}{21}\\ -\frac{1}{3}& \frac{1}{3}\end{array}\right]$

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