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`

Answer

To find the probability that a randomly written `2 x 2` matrix using the elements 1 and -1 is non-singular, we can follow these steps:

### Step 1: Understand the conditions for a non-singular matrix
A `2 x 2` matrix is non-singular if its determinant is non-zero. The determinant of a `2 x 2` matrix

\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

is given by:

\[ \text{det}(A) = ad - bc \]

### Step 2: Identify the possible elements
The elements of the matrix can be either 1 or -1. Therefore, each entry in the matrix can take on one of two values.

### Step 3: Calculate the total number of matrices
Since there are 4 entries in a `2 x 2` matrix and each entry can be either 1 or -1, the total number of possible matrices is:

\[ 2^4 = 16 \]

### Step 4: Identify the cases for non-singular matrices
A `2 x 2` matrix will be non-singular if the determinant \( ad - bc \neq 0 \). This can happen in two specific cases:
1. **Case 1:** One of the entries is -1 and the other three entries are 1.
2. **Case 2:** Three entries are -1 and one entry is 1.

#### Case 1: One -1 and three 1's
The possible matrices are:
- \(\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\)
- \(\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\)
- \(\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}\)
- \(\begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}\)

This gives us **4 matrices**.

#### Case 2: Three -1's and one 1
The possible matrices are:
- \(\begin{pmatrix} 1 & -1 \\ -1 & -1 \end{pmatrix}\)
- \(\begin{pmatrix} -1 & 1 \\ -1 & -1 \end{pmatrix}\)
- \(\begin{pmatrix} -1 & -1 \\ 1 & -1 \end{pmatrix}\)
- \(\begin{pmatrix} -1 & -1 \\ -1 & 1 \end{pmatrix}\)

This also gives us **4 matrices**.

### Step 5: Calculate the total number of non-singular matrices
Adding the two cases together, we find that there are:

\[ 4 \text{ (from Case 1)} + 4 \text{ (from Case 2)} = 8 \text{ non-singular matrices} \]

### Step 6: Calculate the probability
The probability \( P \) that a randomly chosen `2 x 2` matrix is non-singular is given by the ratio of the number of non-singular matrices to the total number of matrices:

\[ P = \frac{\text{Number of non-singular matrices}}{\text{Total number of matrices}} = \frac{8}{16} = \frac{1}{2} \]

### Final Answer
The probability that the matrix is non-singular is \( \frac{1}{2} \).

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