A 2 × 2 square matrix is written down at random using the number 1, -1 as elements. The probability that the matrix is non-singular is

$\frac{1}{2}$

A 2 × 2 square matrix has 4 elements each of which can be chosen in 2 ways.

∴ Total number of 2 × 2 square matrices with elements 1 and -1

$= 2^4 = 16.

Out of these 16 matrices, following matrices are singular:

$\begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix}, \begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix},\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix},\begin{bmatrix} -1 & -1 \\ -1 & -1 \end{bmatrix}$

$\begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & -1 \\ 1 & -1 \end{bmatrix},\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix},\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}$

∴ Number of non-singular matrices = 16 - 8 = 8.

Hence, required probability $=\frac{8}{16}=\frac{1}{2}$