JEE Mathematics 2023 Question with Solution

Given: $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a, b, c, d \in \{0,1,2\}$.

Find: The probability that $M$ is invertible.

The total number of matrices in the sample space is

$$3^4 = 81$$

since each of the four entries can be chosen in $3$ ways.

A $2 \times 2$ matrix is invertible if and only if its determinant is non-zero:

$$\det(M) = ad - bc \neq 0$$

From the calculation, the number of valid matrices satisfying $ad - bc \neq 0$ is $50$.

Therefore,

$$P(A) = \frac{50}{81}$$

So, the correct option is B.

Given: $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with each entry from $\{0,1,2\}$.

Find: $P(A)$ where $A$ is the event that $M$ is invertible.

First count the total number of possible matrices:

$$|S| = 3 \cdot 3 \cdot 3 \cdot 3 = 81$$

Now use the fact that invertibility of a $2 \times 2$ matrix depends on its determinant:

$$\det(M) = ad - bc$$

The matrix is invertible exactly when

$$ad - bc \neq 0$$

The solution states that there are $50$ such matrices for which the determinant is non-zero.

Hence,

$$P(A) = \frac{\text{number of invertible matrices}}{\text{total number of matrices}} = \frac{50}{81}$$

Thus, the required probability is $\frac{50}{81}$ and the correct option is B.