## 1. What does a singular matrix refer to?

### Answer:

### Explanation:

A singular matrix is one that does not have an inverse, which occurs when the determinant of the matrix is zero.

## 2. What is the result of multiplying a matrix by its inverse?

### Answer:

### Explanation:

Multiplying a matrix by its inverse results in the identity matrix.

## 3. What represents the number of linearly independent rows or columns in a matrix?

### Answer:

### Explanation:

The rank of a matrix is defined as the maximum number of linearly independent row vectors (or column vectors) in the matrix.

## 4. In a vector space, what are eigenvectors and eigenvalues associated with?

### Answer:

### Explanation:

Eigenvectors and eigenvalues are concepts associated with a linear transformation represented by a matrix. They represent directions in which the transformation only scales the vectors.

## 5. What is the primary operation performed in Gaussian elimination?

### Answer:

### Explanation:

Gaussian elimination involves the use of elementary row operations to reduce a matrix to row echelon form or reduced row echelon form.

## 6. What is the trace of a matrix?

### Answer:

### Explanation:

The trace of a matrix is defined as the sum of the elements on the main diagonal of the matrix.

## 7. In linear algebra, what is a diagonal matrix?

### Answer:

### Explanation:

A diagonal matrix is a type of matrix where all entries outside the main diagonal are zero.

## 8. What is the determinant of a triangular matrix?

### Answer:

### Explanation:

The determinant of a triangular matrix (upper or lower) is the product of the elements on the main diagonal.

## 9. Which of the following is a property of the transpose of a matrix?

### Answer:

### Explanation:

The transpose of a matrix is achieved by flipping it over its diagonal, so the transpose of a symmetric matrix is the same as the original.

## 10. What does the vector cross product result in?

### Answer:

### Explanation:

The cross product of two vectors results in a vector that is perpendicular to both of the original vectors.

## 11. What is a 'linear combination' of vectors?

### Answer:

### Explanation:

A linear combination involves adding together scalar multiples of vectors.

## 12. What does orthogonality mean in the context of vectors?

### Answer:

### Explanation:

Orthogonality refers to the concept of two vectors being perpendicular, i.e., their dot product is zero.

## 13. What is a subspace in linear algebra?

### Answer:

### Explanation:

A subspace is a subset of a vector space that is also a vector space, containing the zero vector and closed under vector addition and scalar multiplication.

## 14. What is the geometric interpretation of a determinant of a 2×2 matrix?

### Answer:

### Explanation:

The absolute value of the determinant of a 2×2 matrix represents the area of the parallelogram formed by the column vectors of the matrix.

## 15. How many solutions does a system of linear equations have if its coefficient matrix is singular?

### Answer:

### Explanation:

If the coefficient matrix of a system of linear equations is singular, the system has either no solution or infinitely many solutions.

## 16. What is the rank of the identity matrix?

### Answer:

### Explanation:

The rank of the identity matrix is equal to the number of rows (or columns), as it is a full-rank matrix.

## 17. What is a 'vector space'?

### Answer:

### Explanation:

A vector space is a set of vectors along with two operations (vector addition and scalar multiplication) that satisfy certain axioms.

## 18. What does it mean for two matrices to be 'conformable for multiplication'?

### Answer:

### Explanation:

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

## 19. What is the result of the dot product of two orthogonal vectors?

### Answer:

### Explanation:

The dot product of two orthogonal vectors is zero.

## 20. What is a 'basis' of a vector space?

### Answer:

### Explanation:

A basis of a vector space is a set of linearly independent vectors that span the entire vector space.

## 21. What is the main purpose of LU decomposition of a matrix?

### Answer:

### Explanation:

LU decomposition decomposes a matrix into a product of a lower triangular matrix and an upper triangular matrix and is primarily used for solving systems of linear equations.

## 22. How is the length of a vector typically represented?

### Answer:

### Explanation:

The length or magnitude of a vector in Euclidean space is given by the square root of the sum of the squares of its components.

## 23. What is a null space of a matrix?

### Answer:

### Explanation:

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector.

## 24. What is the result of a matrix multiplied by the zero vector?

### Answer:

### Explanation:

Multiplying any matrix by the zero vector always results in the zero vector.

## 25. What does 'linear independence' of a set of vectors imply?

### Answer:

### Explanation:

A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others. This implies that the vectors do not overlap in terms of their span in the vector space.