Determine the rank for the following matrix

1 3
2 4

To determine the rank of the given \( 2 \times 2 \) matrix:

\[
\begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix}
\]

we need to check how many linearly independent rows (or columns) it has. The rank of the matrix is the number of linearly independent rows or columns.

### Step 1: Calculate the determinant of the matrix.

For a \( 2 \times 2 \) matrix of the form:

\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]

The determinant is calculated as:

\[
\text{det}(A) = ad - bc
\]

For the matrix:

\[
\begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix}
\]

The determinant is:

\[
\text{det}(A) = (1 \times 4) - (3 \times 2) = 4 - 6 = -2
\]

### Step 2: Analyze the rank.

- Since the determinant is non-zero, the matrix is of full rank.
- The rank of a \( 2 \times 2 \) matrix with a non-zero determinant is \( 2 \).

Thus, the rank of the matrix is \( \boxed{2} \).

Would you like more details on this process or have any questions?

Here are some related questions:
1. What is the significance of a matrix's determinant?
2. How do we determine the rank of larger matrices?
3. Can a matrix with a zero determinant have a rank greater than 1?
4. What is the geometric interpretation of the rank of a matrix?
5. How does the rank relate to the solution of a system of linear equations?

**Tip:** The rank of a matrix tells you the maximum number of linearly independent rows or columns, which is crucial in understanding the matrix's span and its solutions in linear systems.

Learn how to determine the rank of a 2x2 matrix using the determinant and linear algebra concepts. This example matrix calculation (1, 3; 2, 4) is explained step-by-step, covering the determinant calculation and its implications on matrix rank. Suitable for advanced high school students and beginners in linear algebra.

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