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
Answer & explanation
Correct answer: option 1
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 $=\frac{8}{16}=\frac{1}{2}$