Write the number of all possible matrices of order 2 × 2 with each en

1.	Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.
Answer» We are given with the information that, Each element of the 2 × 2 matrix can be filled in 3 ways, either 1, 2 or 3. We need to find the number of total 2 × 2 matrices with each entry 1, 2 or 3. 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 3 ways, either 1, 2 or 3. That is, Number of ways in which 1 element can be filled = 3¹ Then, Number of ways in which 4 elements can be filled = 3⁴ ⇒ Number of ways in which 4 elements can be filled = 81 Thus, Total number of 2 × 2 matrices with each entry 1, 2 or 3 is 81.

Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.

Answer»

We are given with the information that,

Each element of the 2 × 2 matrix can be filled in 3 ways, either 1, 2 or 3.

We need to find the number of total 2 × 2 matrices with each entry 1, 2 or 3.

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 3 ways, either 1, 2 or 3.

That is,

Number of ways in which 1 element can be filled = 3¹

Then,

Number of ways in which 4 elements can be filled = 3⁴

⇒ Number of ways in which 4 elements can be filled = 81

Thus,

Total number of 2 × 2 matrices with each entry 1, 2 or 3 is 81.

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