Math Problem Statement
Determine the rank for the following matrix
1 3 2 4
Solution
To determine the rank of the given matrix:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Determinant
Formulas
Determinant of a 2x2 matrix: ad - bc
Theorems
-
Suitable Grade Level
Advanced High School