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

• A. \(\frac{1}{9}\)
• B. \(\frac{1}{18}\)
• C. \(\frac{1}{6}\)
• D. \(\frac{1}{12}\)

Hint:

The matrix is not invertible when its determinant ab - 12 is zero.

Answer 1

The Correct Option is A

Step 1: Singular Condition
$\begin{vmatrix}a&2\\6&b\end{vmatrix}=ab-12$, so the matrix is singular exactly when $ab=12$.

Step 2: Check Each Value of a
$a=2\Rightarrow b=6$, $a=3\Rightarrow b=4$, $a=4\Rightarrow b=3$, $a=6\Rightarrow b=2$; $a=1$ and $a=5$ give no valid $b$.

Step 3: Probability
There are $4$ favourable pairs out of $36$, so the probability is $\dfrac{1}{9}$.

Final Answer:
Four of 36 pairs work, giving 1/9. \[ \boxed{\text{(A) }\dfrac{1}{9}} \]