Question

Find the inverse of the matrix

$\left[\begin{array}{cc}2& 4\\ 5& 10\end{array}\right]$

if it exists.

Short answer

The inverse of the given matrix

$\left[\begin{array}{cc}2& 4\\ 5& 10\end{array}\right]$

does not exist.

Step-by-step solution

Step 1. Write down the given equation.

The given matrix is say

$P=\left[\begin{array}{cc}2& 4\\ 5& 10\end{array}\right]$

.

Step 2. Concept used.

The inverse of the matrix$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

exits if $\left|A\right|\ne 0$. And inverse of $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

is given by the formula:

$A^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$where $\left|A\right|\ne 0$

Step 3. Evaluating inverse of given matrix.

The inverse of the matrix is evaluated as:

$\begin{array}{l}P=\left[\begin{array}{cc}2& 4\\ 5& 10\end{array}\right]\mathrm{....}\left(\text{Given}\right)\\ \left|P\right|=20-20\\ =0\end{array}$

Since, $\left|P\right|=0$. Therefore, the inverse of the matrix

role="math" localid="1648235380892" $\left[\begin{array}{cc}2& 4\\ 5& 10\end{array}\right]$ does not exist.

Step 4. Conclusion.

The inverse of the given matrix$\left[\begin{array}{cc}2& 4\\ 5& 10\end{array}\right]$ does not exist.