Consider the following $$2times2$$ matrix A:

$$A=[[5, -1], [2, K]]$$

The value of K for which A has no inverse is most nearly:

A matrix is said to be singular and to lack an inverse if the determinant of the matrix is equal to 0.

To calculate the determinant of a general 2x2 square matrix, we use the following formula:

$\left.openM\right|=ad-bc, where \left.openM\right| represents the determinant of matrix M = \left.open\begin{array}{cc}a& b\\ c& d\end{array}\right]$

For our problem, we want to find what value of K forces the determinant of matrix A to be equal to 0. We use the above formula in order to set up our equation:

$\left.open\mathit{A}\right|=\left(5\right)\left(K\right)-(-1)\left(2\right)$

$0 = \left(5\right)\left(K\right) - (-1)\left(2\right)$

$0 = 5K + 2$

$K = \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

Therefor, the value of K for which A has no inverse is most nearly -2/5.