Counting Distinct Matrices with Four Entries from (1, 2)

The question asks for the number of distinct matrices that can be formed using exactly four entries, where each entry must be either 1 or 2. A matrix is defined by its entries and their positions. When we talk about matrices with four entries, there are typically three common shapes (dimensions) that have exactly four entries:

• A matrix with 1 row and 4 columns (1x4 matrix).
• A matrix with 4 rows and 1 column (4x1 matrix).
• A matrix with 2 rows and 2 columns (2x2 matrix).

For a matrix to be distinct, its dimensions or any of its entries must be different from another matrix.

Counting Distinct Matrices for Each Shape

Let's consider each possible shape with four entries:

1. 1x4 Matrix

A 1x4 matrix has 1 row and 4 columns. It looks like this:

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

There are 4 entry positions (a, b, c, d). For each position, the entry must be taken from the set (1, 2). This means there are 2 choices for entry 'a', 2 choices for entry 'b', 2 choices for entry 'c', and 2 choices for entry 'd'.

The total number of distinct 1x4 matrices is the product of the number of choices for each position:

Number of distinct 1x4 matrices = $2 \times 2 \times 2 \times 2 = 2^4 = 16$.

2. 4x1 Matrix

A 4x1 matrix has 4 rows and 1 column. It looks like this:

$\begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix}$

Similar to the 1x4 matrix, there are 4 entry positions. For each position, the entry must be taken from the set (1, 2), giving 2 choices per position.

The total number of distinct 4x1 matrices is the product of the number of choices for each position:

Number of distinct 4x1 matrices = $2 \times 2 \times 2 \times 2 = 2^4 = 16$.

3. 2x2 Matrix

A 2x2 matrix has 2 rows and 2 columns. It looks like this:

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

Again, there are 4 entry positions. For each position, the entry must be taken from the set (1, 2), giving 2 choices per position.

The total number of distinct 2x2 matrices is the product of the number of choices for each position:

Number of distinct 2x2 matrices = $2 \times 2 \times 2 \times 2 = 2^4 = 16$.

Total Number of Distinct Matrices

Since the question asks for the total number of distinct matrices with four entries taken from (1, 2), and these matrices can have different shapes (1x4, 4x1, or 2x2), we sum the number of distinct matrices for each shape:

Total distinct matrices = (Distinct 1x4 matrices) + (Distinct 4x1 matrices) + (Distinct 2x2 matrices)

Total distinct matrices = $16 + 16 + 16 = 48$.

Therefore, there are 48 distinct matrices with all four entries taken from (1, 2), considering the common matrix shapes with four entries.

Revision Table: Counting Matrices

Additional Information: Combinatorics and Matrices

This problem involves basic combinatorics, specifically the multiplication principle. The multiplication principle states that if there are 'n' ways to do one thing and 'm' ways to do another, then there are 'n * m' ways to do both. In this case, placing an entry in each position is an independent event, and there are 2 ways to choose the entry for each of the 4 positions.

When dealing with matrix problems, it's crucial to pay attention to:

• Dimensions/Shape: A matrix's properties often depend heavily on its number of rows and columns.
• Entry Set: The set from which entries are drawn determines the possible values for each position.
• Distinct Matrices: Two matrices are distinct if they differ in shape or if they have the same shape but at least one entry in a corresponding position is different.

In more complex matrix counting problems, constraints might be placed on the sum of entries, the determinant, the rank, or other matrix properties, requiring more advanced combinatorial techniques.