A square matrix is singular if and only if its determinant is:

Linear Algebra Quiz

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Mathematics
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University
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Hard
Cường Vũ
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67 questions
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1.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
1
-1
0
any non-zero value
Answer explanation
A square matrix is singular if its determinant is 0. This means it does not have an inverse. Therefore, the correct answer is 0, as singular matrices cannot have non-zero determinants.
2.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
If the reduced row echelon form of the augmented matrix of a system has a row of zeros followed by a non-zero entry, then the system is:
consistent
inconsistent
dependent
underdetermined
Answer explanation
A row of zeros followed by a non-zero entry in the reduced row echelon form indicates a contradiction, meaning the system has no solutions. Therefore, the system is inconsistent.
3.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
The trace of a matrix is the sum of its:
all entries
diagonal entries
eigenvalues
entries in the first row
Answer explanation
The trace of a matrix is defined as the sum of its diagonal entries. This means you only consider the elements that run from the top left to the bottom right of the matrix, making 'diagonal entries' the correct choice.
4.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
What condition must be satisfied for Cramer's Rule to be applicable to a system of linear equations?
The system must be homogeneous.
The determinant of the coefficient matrix must be zero.
The determinant of the coefficient matrix must be nonzero.
The number of equations must equal the number of unknowns.
Answer explanation
Cramer's Rule applies when the determinant of the coefficient matrix is nonzero, ensuring a unique solution exists for the system of linear equations. This condition is essential for the rule to be valid.
5.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
What is the dimension of the solution space for Ax = 0, given A = [[5, 5, 4, 4], [3, 1, -2, 3], [7, 7, 10, 7], [2, 7, 4, -4]]? (Requires row reduction to find the rank)
0
1
2
3
Answer explanation
To find the dimension of the solution space for Ax = 0, we row reduce A to find its rank. The rank is 3, so the nullity (dimension of the solution space) is 4 - 3 = 1. Thus, the correct answer is 1.
6.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
Consider the system: 4x - 5y + 7z = 22; 5x - 6y + 8z = 7; 6x - 7y + 9z = -8; 7x - 8y + 10z = -23. Is this system consistent?
Yes
No
Cannot be determined.
Only if the determinant is zero.
Answer explanation
The system is inconsistent because the equations represent planes that do not intersect at a common point. Therefore, the correct answer is 'No'.
7.
MULTIPLE CHOICE QUESTION
1 min • 1 pt
For the system in Question 2, if it were consistent, what would be the form of a general solution (xₚ + xh)?
x = xₚ + xh where xₚ is a particular solution and xh is a solution to the homogeneous system.
x = xₚ - xh where xₚ is a particular solution and xh is a solution to the homogeneous system.
x = xₚ + cxh where xₚ is a particular solution, xh is a solution to the homogeneous system, and c is a scalar.
No general solution exists for inconsistent systems.
Answer explanation
The general solution for a consistent system is given by x = xₚ + xₕ, where xₚ is a particular solution and xₕ is a solution to the homogeneous system. This reflects the superposition principle in linear systems.
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