Linear Algebra Quiz

This linear algebra quiz tests your knowledge on some of the fundamental principles that are crucial in the field of linear algebra. A holistic understanding in linear algebra is required to succeed on this quiz

The quiz covers many of the concepts taught in first year university courses of linear algebra. Take the quiz to get a sense of how well you understand linear algebra and what areas you need to improve on!

Created by: Yihao Xu

The determinant of any identity matrix is:
- 0
- 1
- -1
- Cannot be determined
A matrix is not invertible if:
- Its determinant is not zero
- Its columns form a linearly independent set
- Its transpose is not invertible
- It is a square matrix
- Its eigenvalues does not include zero
- More than one of the above
A system of m linear equations in n variables is inconsistent if:
- m is less than n
- m is greater than n
- The system's corresponding coefficient matrix has a rank less than m
- The system's corresponding augmented matrix has a rank greater than n
- Two of the above
- Three of the above
- All of the above
- None of the above
A system of m linear equations in n variables cannot have a unique solution if:
- m is less than n
- m is greater than n
- The system's corresponding coefficient matrix has a rank less than m
- The system's corresponding augmented matrix has a rank of n
- Two of the above
- Three of the above
- All of the above
- None of the above
A 4x4 matrix cannot be an elementary matrix if:
- Its entry at row 2 and column 2 is 0
- Its entry at row 1 and column 4 is 3
- Its entry at row 3 and column 3 is -1/2
- There are more than 2 non-zero entries in column 2
- It has the same number of non-zero entries as the 4x4 identity matrix
- Its determinant is less than 0
Multiplying an nxn elementary matrix on the right of an mxn matrix A is equivalent to:
- Performing the corresponding row operation on A
- Performing the corresponding row operation on the transpose of A
- Performing the corresponding column operation on A
- Performing the corresponding column operation on the transpose of A
- Its behavior cannot be characterized
A set of k vectors in n dimension is a basis for R^n if:
- k is less than or equal to n and the set is linearly independent
- k is greater than or equal to n and the set is linearly independent
- k equals n
- The set spans R^n
- None of the above
An idempotent matrix:
- Is always diagonalizable
- Has trace equal to its rank
- Has only two distinct eigenvalues
- Is invertible if and only if it is the identity matrix
- Two of the above
- Three of the above
- All of the above
The number of free variables in a system of m linear equations in n variables depends on:
- m and n
- m and the rank of the system's corresponding coefficient matrix
- n and the rank of the system's corresponding coefficient matrix
- m and the rank of the system's corresponding augmented matrix
- n and the rank of the system's corresponding augmented matrix
- m, n, and the rank of the system's corresponding coefficient matrix
- m, n, and the rank of the system's corresponding augmented matrix
Choose the statement that is true
- Every diagonalizable matrix is invertible.
- Every invertible matrix is diagonalizable.
- There does not exist an algorithm to compute powers of a matrix that is not diagonalizable
- Every real matrix with real eigenvalues is triangulable
- None of the above

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