What is the total number of 2 × 2 matrices with each entry 0 or 1?

1.	What is the total number of 2 × 2 matrices with each entry 0 or 1?
Answer» We are given with the information that, Each element of the 2 × 2 matrix can be filled in 2 ways, either 0 or 1. We need to find the number of total 2 × 2 matrices with each entry 0 or 1. Let A be 2 × 2 matrix such that, A = \( \begin{bmatrix}a_{11} & a_{12} \\[0.3em]a_{21} & a_{22}\\[0.3em]\end{bmatrix}\) Note that, There are 4 elements in the matrix. So, If 1 element can be filled in 2 ways, either 0 or 1. That is, Number of ways in which 1 element can be filled = 2¹ Then, Number of ways in which 4 elements can be filled = 2⁴ ⇒ Number of ways in which 4 elements can be filled = 16 Thus, Total number of 2 × 2 matrices with each entry 0 or 1 is 16.

Answer»

We are given with the information that,

Each element of the 2 × 2 matrix can be filled in 2 ways, either 0 or 1.

We need to find the number of total 2 × 2 matrices with each entry 0 or 1.

Let A be 2 × 2 matrix such that,

A = \( \begin{bmatrix}a_{11} & a_{12} \\[0.3em]a_{21} & a_{22}\\[0.3em]\end{bmatrix}\)

Note that,

There are 4 elements in the matrix.

So,

If 1 element can be filled in 2 ways, either 0 or 1.

That is,

Number of ways in which 1 element can be filled = 2¹

Then,

Number of ways in which 4 elements can be filled = 2⁴

⇒ Number of ways in which 4 elements can be filled = 16

Thus,

Total number of 2 × 2 matrices with each entry 0 or 1 is 16.

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