Linear Algebra Quiz

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

• 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.

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.

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.

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.

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.

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'.

For the system in Question 2, if it were consistent, what would be the form of a general solution (xₚ + x_h)?

• x = xₚ + x_h where xₚ is a particular solution and x_h is a solution to the homogeneous system.

• x = xₚ - x_h where xₚ is a particular solution and x_h is a solution to the homogeneous system.

• x = xₚ + cx_h where xₚ is a particular solution, x_h 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.

Given the matrix [[2, 6, -2], [3, p, 5], [7, 14, 2p-3]], for what value of 'p' does the matrix have rank 2? (Requires row reduction to find the condition for rank 2)

• p = 3

• p = 4

• p = 5

• p = 6

Answer explanation

To find the rank, we row reduce the matrix. Setting p = 3 leads to a row of zeros, resulting in a rank of 2. Other values of p do not yield a rank of 2. Thus, the correct value is p = 3.

For the matrix in Question 4, what values of 'p' result in a rank of 3?

• p = 3

• p ≠ 3

• p = 0

• p ≠ 0

Answer explanation

For the matrix to have a rank of 3, it must not lose a row of linear independence. If p = 3, the matrix may become dependent, reducing the rank. Thus, p must not equal 3, making p ≠ 3 the correct choice.

Consider the system: x + y + z = 2; -2x - 3y - z = 4; 3x + 4y + 2z = p. For what value of 'p' does this system have infinitely many solutions? (Requires Gaussian elimination or row reduction)

• p = 0

• p = 10

• p = 12

• p = 14

Answer explanation

To have infinitely many solutions, the third equation must be a linear combination of the first two. Row reducing the first two equations shows that for p = 10, the third equation aligns, confirming infinitely many solutions.