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

Options

A. `1//2`
B. `3//8`
C. `5//8`
D. `1//3`

Explanation

To find the probability that a randomly formed \(2 \times 2\) matrix using the elements \(1\) and \(-1\) is non-singular, we can follow these steps:

### Step 1: Determine the total number of matrices
A \(2 \times 2\) matrix can be represented as:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
where each of the elements \(a\), \(b\), \(c\), and \(d\) can be either \(1\) or \(-1\). Since there are 4 elements and each can take 2 values, the total number of matrices that can be formed is:
\[ 2^4 = 16 \]

### Step 2: Identify the conditions for a matrix to be singular
A \(2 \times 2\) matrix is singular if its determinant is zero. The determinant of matrix \(A\) is given by:
\[ \text{det}(A) = ad - bc \]
We need to find the combinations of \(a\), \(b\), \(c\), and \(d\) that make this determinant equal to zero.

### Step 3: List the singular matrices
We can find the singular matrices by checking combinations where \(ad = bc\). The following matrices are singular:
1. \(\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\)
2. \(\begin{pmatrix} -1 & -1 \\ -1 & -1 \end{pmatrix}\)
3. \(\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}\)
4. \(\begin{pmatrix} -1 & 1 \\ -1 & 1 \end{pmatrix}\)
5. \(\begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}\)
6. \(\begin{pmatrix} -1 & -1 \\ 1 & 1 \end{pmatrix}\)
7. \(\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}\)
8. \(\begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}\)

In total, there are 8 singular matrices.

### Step 4: Calculate the number of non-singular matrices
The number of non-singular matrices is the total number of matrices minus the number of singular matrices:
\[ \text{Number of non-singular matrices} = 16 - 8 = 8 \]

### Step 5: Calculate the probability
The probability that a randomly formed \(2 \times 2\) matrix is non-singular is given by the ratio of non-singular matrices to the total number of matrices:
\[ P(\text{non-singular}) = \frac{\text{Number of non-singular matrices}}{\text{Total number of matrices}} = \frac{8}{16} = \frac{1}{2} \]

Thus, the probability that the matrix is non-singular is \(\frac{1}{2}\).