The question in the image asks:

"The number of all possible matrices of order 3x3 with each entry being either 0 or -1 is:"

Solution:

A 3x3 matrix has $9$ elements. Each element of the matrix can take two values, either $0$ or $-1$.

Thus, the total number of possible matrices is the total number of ways to fill 9 positions where each position has 2 choices (0 or -1). This is calculated as:

$2^9 = 512$

Final Answer:

The number of all possible matrices is 512, which corresponds to the first option in the image.

Would you like a detailed explanation of this calculation? Or have any questions about similar matrix problems?

Here are 5 related questions to expand your understanding:

1. What is the total number of 2x2 matrices with entries restricted to 0, 1, and -1?
2. How many different matrices can be formed if the entries are restricted to $0$, $1$, and $2$ for a 2x3 matrix?
3. What is the number of 4x4 matrices if each entry can be either $0$ or $1$?
4. Can you calculate the number of matrices of order $n \times n$ with each entry either $0$ or $1$?
5. What happens to the number of possible matrices if the entries can be real numbers?

Tip: Whenever dealing with matrices with restricted entries, you can always find the number of possible matrices by raising the number of choices per entry to the power of the number of elements in the matrix.