Question:
Let \(a\) and \(b\) be the numbers obtained in two trials of tossing a fair die. The probability that the matrix \(\begin{bmatrix} a & 2 \\ 6 & b \end{bmatrix}\) is not invertible is
Let \(a\) and \(b\) be the numbers obtained in two trials of tossing a fair die. The probability that the matrix \(\begin{bmatrix} a & 2 \\ 6 & b \end{bmatrix}\) is not invertible is
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The matrix is not invertible when its determinant ab - 12 is zero.
Updated On: Nov 12, 2026
- \(\frac{1}{9}\)
- \(\frac{1}{18}\)
- \(\frac{1}{6}\)
- \(\frac{1}{12}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
A matrix is not invertible when its determinant is zero: \(ab-12=0\), i.e. \(ab=12\).
Step 2: Count Favourable Outcomes
With \(a,b\in\{1,\dots,6\}\), the pairs with \(ab=12\) are \((2,6),(3,4),(4,3),(6,2)\). That is \(4\) outcomes.
Step 3: Total Outcomes
Two tosses of a die give \(6\times6=36\) outcomes.
Step 4: Probability
\(P=\dfrac{4}{36}=\dfrac{1}{9}\), option (A).
Final Answer:
The probability is 1/9. \[ \boxed{\text{(A) }\dfrac{1}{9}} \]
A matrix is not invertible when its determinant is zero: \(ab-12=0\), i.e. \(ab=12\).
Step 2: Count Favourable Outcomes
With \(a,b\in\{1,\dots,6\}\), the pairs with \(ab=12\) are \((2,6),(3,4),(4,3),(6,2)\). That is \(4\) outcomes.
Step 3: Total Outcomes
Two tosses of a die give \(6\times6=36\) outcomes.
Step 4: Probability
\(P=\dfrac{4}{36}=\dfrac{1}{9}\), option (A).
Final Answer:
The probability is 1/9. \[ \boxed{\text{(A) }\dfrac{1}{9}} \]
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