Question

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 and 1 only. The probability that the determinant chosen is non-zero is

• A. 5/16
• B. 3/8
• C. 1/4
• D. 5/8

Answer 1

The Correct Option is A

Given: We are tasked with finding the probability that a randomly chosen 2x2 determinant with elements 0 and 1 only is non-zero. Step 1: Understand the possible 2x2 matrices. A 2x2 matrix has 4 elements. Since each element can either be 0 or 1, there are a total of \( 2^4 = 16 \) possible matrices. The general form of a 2x2 matrix is: $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ where \( a, b, c, d \in \{0, 1\} \). Step 2: Calculate the determinant of the matrix. The determinant of a 2x2 matrix is given by: $$ \text{det}\left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = ad - bc $$ To determine when the determinant is non-zero, we need to consider the cases when \( ad - bc \neq 0 \). Step 3: Analyze when the determinant is non-zero. 1. **Case 1: \( a = 0 \) and \( b = 0 \):** - The determinant is \( 0 \cdot d - 0 \cdot c = 0 \). 2. **Case 2: \( a = 0 \) and \( b = 1 \):** - The determinant is \( 0 \cdot d - 1 \cdot c = -c \), which is \( 0 \) if \( c = 0 \) and \( -1 \) if \( c = 1 \), so it can be non-zero. 3. **Case 3: \( a = 1 \) and \( b = 0 \):** - The determinant is \( 1 \cdot d - 0 \cdot c = d \), which is \( 0 \) if \( d = 0 \) and \( 1 \) if \( d = 1 \), so it can be non-zero. 4. **Case 4: \( a = 1 \) and \( b = 1 \):** - The determinant is \( 1 \cdot d - 1 \cdot c = d - c \), which is \( 0 \) if \( d = c \) and non-zero if \( d \neq c \). Step 4: Count the number of matrices with non-zero determinant. Now, let's count how many matrices result in a non-zero determinant: - For \( a = 0 \) and \( b = 0 \), the determinant is always 0 (1 case). - For \( a = 0 \) and \( b = 1 \), the determinant is non-zero if \( c = 0 \) and \( d = 1 \), or \( c = 1 \) and \( d = 0 \) (2 cases). - For \( a = 1 \) and \( b = 0 \), the determinant is non-zero if \( d = 1 \) (1 case). - For \( a = 1 \) and \( b = 1 \), the determinant is non-zero if \( d \neq c \) (2 cases). Thus, the total number of matrices with a non-zero determinant is \( 2 + 1 + 2 = 5 \). Step 5: Calculate the probability. Since there are 16 total possible matrices, the probability of choosing a matrix with a non-zero determinant is: $$ \frac{5}{16} $$ Conclusion: The probability that the determinant chosen is non-zero is \( \boxed{\frac{5}{16}} \).