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The document you uploaded contains a **Mid-term Assignment** for the course **MAT217** from Southeast University, covering **Linear Algebra, Coordinate Geometry, and Vector Analysis**. Here’s an outline of the problems:

1. **Matrix Operations**:
   - (a) Find the determinant of a matrix \( B \).
   - (b) Find the inverse of matrix \( A \) using Echelon form.
   - (c) Calculate the rank of matrix \( A \) by reducing it to Echelon form.
   - (d) Express a given vector as a linear combination of other vectors in \( \mathbb{R}^4 \).

2. **Vector Operations in \( \mathbb{R}^3 \)**:
   - (a) Find the magnitude of a vector expression involving \( a \), \( b \), and \( c \).
   - (b) Determine the angle between vectors \( a \) and \( c \).
   - (c) Find a unit vector perpendicular to \( a \times b \).
   - (d) Compute the dot and cross products for various vector combinations.

Would you like help solving any of these problems in particular? Feel free to specify.

Here are some follow-up questions:
1. How can we find the determinant of a matrix using cofactor expansion?
2. What is the process for finding the inverse of a matrix using row reduction?
3. How do you calculate the rank of a matrix through echelon form?
4. What is the method for expressing a vector as a linear combination in \( \mathbb{R}^n \)?
5. How do we calculate the angle between two vectors using the dot product?

**Tip**: When finding the inverse of a matrix, reducing it to row echelon form step-by-step ensures accuracy.

Solve all problems in the MAT217 Mid-term Assignment, Summer 2024, including matrix operations (determinants, inverses, rank) and vector operations (magnitudes, angles, cross products, and linear combinations) from Linear Algebra, Coordinate Geometry, and Vector Analysis.

Solve mid-term assignment problems from MAT217, covering matrix determinants, inverses, rank, and vector operations such as magnitudes, angles, and cross products. This guide is suitable for undergraduate students studying Linear Algebra, Coordinate Geometry, and Vector Analysis.

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