JEE Mathematics 2025 Question with Solution

Given: A matrix of order $2$ has entries from the set $\{2, 3, 6, 9\}$.

Find: The number of singular matrices.

For a general $2 \times 2$ matrix,

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

it is singular when its determinant is zero:

$$ad - bc = 0$$

So we need

$$ad = bc$$

Each of $a, b, c, d$ is chosen from $\{2, 3, 6, 9\}$.

The total number of possible matrices is

$$4^4 = 256$$

Now count those arrangements for which $ad = bc$.

Using enumeration over all $4$-tuples $(a, b, c, d) \in \{2, 3, 6, 9\}^4$, the number of matrices satisfying the singularity condition is found to be

$$36$$

Therefore, the number of singular matrices is $36$.

Given: The entries of a singular matrix of order $2$ are chosen from $\{2, 3, 6, 9\}$.

Find: The number of such singular matrices.

The determinant condition is

$$ad = bc$$

A casewise count gives the total.

1. Case I: Exactly one number is used

All entries are equal, so every such matrix is singular.

$${}^4C_1 = 4$$

2. Case II: Exactly two numbers are used

The working states that after applying the condition $ad = bc$, the number of singular matrices in this case is

$$24$$

3. Case III: Exactly three numbers are used

No singular matrix occurs in this case.

$$0$$

4. Case IV: Exactly four numbers are used

Use the product relation

$$2 \times 9 = 3 \times 6$$

Hence this case contributes

$$8$$

Adding all contributions,

$$4 + 24 + 0 + 8 = 36$$

Therefore, the number of singular matrices is $36$.

Note: The second approach in the source has inconsistent matrix notation in one line, but its final determinant condition and total count agree with the answer $36$.