WorksheetsLinear Algebra Quiz
Total questions: 142
Worksheet time: 2hrs 22mins
Let u = (1, 2, 3) and v = (4, 5, 6). Compute 2u - 3v.
(-10, -11, -12)
(10, 11, 12)
(-10, -11, 12)
(10, 11, -12)
Determine if the vectors u = (1, 2) and v = (3, 4) are linearly independent.
Yes
No
Only if they are orthogonal.
Only if they are normalized.
Find the value of k such that the vectors u = (1, k) and v = (2, 3) are linearly dependent.
k = 3/2
k = 2/3
k = 6
k = 0
Given the vectors u = (1, 2, 3) and v = (4, 0, -1), compute their dot product u ⋅ v.
1
2
3
4
What is the length (magnitude) of the vector v = (3, 4)?
3
4
5
7
Normalize the vector u = (1, 1).
(1/√2, 1/√2)
(1, 1)
(√2, √2)
(0, 0)
Are the vectors u = (1, 2, 3), v = (4, 5, 6), and w = (7, 8, 9) linearly independent?
Yes
No
It depends on the scalar multiples
It is impossible to determine without further information
Find scalars a and b such that a(1, 2) + b(3, 1) = (0, 0) (assuming a and b are not both zero).
a = 0, b = 0
a = 1, b = -2
a = 2, b = -1
a = 3, b = -1
Determine if the set of vectors S = {(1, 0, 0), (0, 1, 0), (1, 1, 0)} spans R³.
Yes
No
Only if the components are integers.
Only if the components are positive.
What is the dimension of the vector space P₂ (polynomials of degree ≤ 2)?
1
2
3
Infinite
If A is a 3x5 matrix and B is a 5x2 matrix, what is the size of AB?
3x5
5x2
3x2
AB is undefined.
Find the matrix product AB, given A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
[[19, 22], [43, 50]]
[[12, 14], [21, 32]]
[[5, 12], [21, 32]]
[[19, 22], [23, 26]]
Calculate the determinant of the matrix: [[2, 0, 1], [0, 1, 0], [1, 0, 2]]
3
0
1
-3
If A = [[1, 2], [3, 4]] and B = [[5, 0], [0, 2]], compute AB.
[[5, 4], [15, 8]]
[[1, 4], [3, 8]]
[[5, 0], [15, 8]]
[[1, 0], [3, 0]]
Find the value of k such that the vectors (1, k) and (2, 6) are linearly dependent.
k = 3
k = 12
k = 0
No such value of k exists.
What is the dot product of the vectors u = (2, -1, 3) and v = (4, 1, 0)?
7
9
11
14
Given the vectors u = (1, 2) and v = (3, 4), find a linear combination au + bv that equals (0, 0) (where a and b are not both zero).
a = 0, b = 0
a = 3, b = -1
a = 2, b = -1
a = 4, b = -3
If A is a 3x3 matrix and det(A) = 5, what is det(3A)?
5
15
45
125
Determine if the set S = {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)} is linearly independent in R³.
Yes
No
It depends on scalar multiples
More information is needed
If a square matrix has a row of all zeros, what is its determinant?
1
-1
It depends on the other rows.
0
What is the size of the transpose of a 5x3 matrix?
5x3
3x5
5x5
3x3
If A is an invertible 3x3 matrix and det(A) = 4, what is det(A⁻¹)?
4
1/4
-4
0
Find the determinant of the matrix: [[1, 2, 3], [0, 1, 4], [0, 0, 1]]
0
1
6
10
Two vectors in R² are linearly dependent if and only if:
ix: [[1, 2, 3], [0, 1, 4], [0, 0, 1]]
0
1
6
10
Two vectors in R² are linearly dependent if and only if:
They are orthogonal.
One is a scalar multiple of the other.
Their dot product is zero.
Their lengths are equal.
Is the set of vectors S = {(1, 2, 3), (2, 4, 6)} linearly independent?
Yes
No
It depends on the scalar multiples.
Cannot be determined.
What is the dimension of the vector space of all 2x2 matrices?
1
2
4
Infinite
If A is a 3x3 matrix and det(A) = 0, then the system Ax = 0:
has only the trivial solution (x = 0).
has a unique non-trivial solution.
has infinitely many solutions.
has no solution.
If you add a multiple of one row of a matrix to another row, what happens to the determinant?
It is multiplied by that multiple.
It changes sign.
It remains unchanged.
It becomes zero.
If the determinant of the coefficient matrix of a system of linear equations is zero, what can be said about the system?
It has a unique solution.
It has no solution.
It has infinitely many solutions or no solution.
It has infinitely many solutions.
If A is a square matrix and Aᵀ = A, then A is:
Skew-symmetric
Orthogonal
Symmetric
Nilpotent
Find the determinant of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
1
0
-1
10
Calculate the determinant of the matrix B = [[2, -1, 0], [1, 2, 3], [0, 1, -1]].
13
-13
11
-11
Given a 3x3 matrix A with det(A) = 7, find det(4A).
7
28
112
343
If A is a 2x2 matrix and det(A) = 5, what is det(Aᵀ)?
5
-5
1/5
0
If you swap two columns of a square matrix, what happens to its determinant?
It remains the same.
It is multiplied by 2.
It changes its sign.
It becomes zero.
If you multiply a single row of a matrix by a scalar k, the determinant is multiplied by:
1/k
k
k²
k³
A square matrix is singular if its determinant is:
1
-1
0
any non-zero number
If A and B are both invertible nxn matrices, then (AB)⁻¹ is equal to:
A⁻¹B⁻¹
B⁻¹A⁻¹
A⁻¹ + B⁻¹
AB
If A is an idempotent matrix (A² = A) and A is invertible, then A must be:
the zero matrix
the identity matrix
a nilpotent matrix
a skew-symmetric matrix
Find the inverse of the matrix A = [[2, 0, 0], [0, -1, 0], [0, 0, 3]].
[[1/2, 0, 0], [0, -1, 0], [0, 0, 1/3]]
[[2, 0, 0], [0, -1, 0], [0, 0, 3]]
[[0, 0, 1/3], [0, -1, 0], [1/2, 0, 0]]
The inverse doesn't exist.
A homogeneous system of linear equations Ax = 0 always has:
a unique solution
infinitely many solutions
at least the trivial solution (x = 0)
no solution
If a homogeneous system of linear equations has a non-trivial solution, what is true about the determinant of its coefficient matrix?
It is 1.
It is -1.
It is nonzero.
It is 0.
Let u = (1, 2) and v = (3, 4). Calculate the dot product u ⋅ v.
10
11
12
13
What is the length (magnitude) of the vector w = (-3, 4)?
1
5
7
12
Normalize the vector v = (2, 2).
(1, 1)
(1/√2, 1/√2)
(√2, √2)
(2, 2)
What is the length (magnitude) of the vector w = (-3, 4)?
1
5
7
12
Normalize the vector v = (2, 2).
(1, 1)
(1/√2, 1/√2)
(√2, √2)
(2, 2)
Find the value of k such that the vectors (1, k) and (2, 6) are linearly dependent.
k = 3
k = 12
k = 0
No such k exists.
If the determinant of a 3x3 matrix A is 0, what can be said about the columns of A?
They are linearly independent.
They are linearly dependent.
They are orthogonal.
They are all zero vectors.
Solve the system of equations using matrices: x + y = 3; 2x - y = 0
x = 1, y = 2
x = 2, y = 1
x = 0, y = 3
No solution.
If A is a 3x3 matrix and det(A) = -5, find det(-A).
5
-5
1/5
0
If a system of linear equations has more variables than equations, what is true about its solutions?
It has a unique solution.
It has no solution.
It has infinitely many solutions or no solution.
It has exactly two solutions.
If the reduced row echelon form of an augmented matrix has a row of zeros followed by a non-zero entry, what can be said about the system?
It has infinitely many solutions.
It has no solution.
It has a unique solution.
The system is homogeneous.
The trace of a matrix is the sum of its:
All entries
Diagonal entries
Eigenvalues
Entries in the first row
If the determinant of the coefficient matrix of a system of linear equations is non-zero, the system has:
No solution
Infinitely many solutions
A unique solution
At least two solutions
If two rows of a matrix are equal, its determinant is:
1
-1
2
0
If a square matrix A satisfies A² = I (identity matrix), then A is called:
Idempotent
Nilpotent
Involutory
Orthogonal
What is the dimension of the vector space P₃ (polynomials of degree ≤ 3)?
1
2
3
4
If A is an invertible matrix, then det(Aᵀ) is:
0
1
det(A)
-det(A)
If a matrix is invertible, then its determinant must be:
1
0
-1
Non-zero
If the augmented matrix of a system of linear equations reduces to [[1, 0, 2; 0, 1, 3; 0, 0, 0]], how many solutions does the system have?
0
1
Infinitely many
2
A matrix is called skew-symmetric if:
Aᵀ = A
Aᵀ = -A
A² = A
A² = I
Let A be a 3x3 matrix with det(A) = 7. What is det(2A⁻¹)?
14
7/2
2/7
1/14
Find the determinant of the matrix: [[1, 2, 3], [0, 1, 4], [5, 0, 6]] using cofactor expansion along the first row.
22
-2
2
0
Calculate the determinant of the matrix: [[2, 0, 1, 0], [0, 1, 0, 2], [1, 0, 2, 0], [0, 2, 0, 1]].
9
-9
7
-7
If A and B are 4x4 matrices with det(A) = 2 and det(B) = 3, what is det(ABᵀ)?
6
1/6
5
1
If matrix A is 2x3 and matrix B is 3x1, what size is the resulting matrix AB?
2x1
3x3
2x3
3x1
Given A = [[1, 2], [3, 4]] and A⁻¹ = [[-2, 1], [3/2, -1/2]], verify AA⁻¹ = I. Which matrix results?
[[1, 0], [0, 1]]
[[0, 1], [1, 0]]
[[1, 1], [0, 1]]
[[0, 0], [0, 0]]
If you interchange two rows of a 5x5 matrix, its determinant is multiplied by:
1
2
-1
5
If you interchange two rows of a 5x5 matrix, its determinant is multiplied by:
1
2
-1
5
If a matrix has linearly dependent rows, its determinant is:
1
-1
any non-zero number
0
Solve for x: det([[x, 2], [3, 1]]) = 4
x = 6
x = 0
x = -6
x = 1
If A is a 3x3 matrix and det(A) = -2, what is det(5A)?
-10
-50
-250
-1000
If A is an invertible matrix, then det(A⁻¹) = ?
det(A)
1/det(A)
-det(A)
0
Given the system of equations: x + 2y = 5; 2x - y = 1. What is the value of x?
1
2
3
4
If A is a singular matrix (not invertible), what is its determinant?
1
-1
Any non-zero number
0
What is the trace of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
15
18
0
27
A matrix is idempotent if:
A² = 0
A² = I
A² = A
Aᵀ = A
If A is a 3x3 matrix and det(A) = 5, what is det(Aᵀ)?
5
-5
1/5
0
If two rows of a matrix are linearly dependent, what is the value of its determinant?
1
-1
Any non-zero number
0
A system of linear equations is overdetermined if:
it has more variables than equations
it has more equations than variables
the determinant is 0
it has a unique solution
If A is a 2x2 matrix such that A² = 0 (zero matrix), then det(A) = ?
1
-1
2
0
What is the size of the adjoint of a 5x5 matrix?
1x1
5x5
25x25
25x1
If A is a square matrix and det(A) ≠ 0, what can be said about A?
A is singular
A is invertible
A is the zero matrix
A is the identity matrix
A homogeneous system of linear equations always has:
exactly one solution
infinitely many solutions
no solution
at least the trivial solution (x = 0)
If the determinant of a coefficient matrix is zero, what can be said about the system of equations?
It has a unique solution.
It has infinitely many solutions.
It has no solution.
It has no solution or infinitely many solutions.
If A is an n x n matrix and rank(A) < n, what can you say about A?
It is invertible.
It is not invertible.
It is a diagonal matrix.
It is an identity matrix.
If A is an orthogonal matrix (AᵀA = I), what are the possible values of det(A)?
0
1
-1
1 or -1
Find the value of k such that the matrix [[k, 2], [3, 4]] is singular (non-invertible).
k = 3/2
k = 2/3
k = 6
k = 0
What is the dimension of the vector space of all 3x3 matrices?
3
6
9
Infinite
If A is a 2x2 matrix and Aᵀ = -A, then A is called:
symmetric
orthogonal
skew-symmetric
idempotent
What operation does NOT preserve the determinant of a matrix?
Swapping two rows
Multiplying a row by a scalar
Adding a multiple of one row to another
Taking the transpose
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
Find the determinant of the matrix A = [[2, -1, 3], [1, 0, 4], [0, 2, 1]].
11
-11
13
-13
Find the determinant of the matrix A = [[2, -1, 3], [1, 0, 4], [0, 2, 1]].
11
-11
13
-13
Given a 3x3 matrix A with det(A) = 5, find det(3Aᵀ).
5
15
45
135
If you multiply a single column of a matrix by -2, what happens to its determinant?
It's multiplied by -2.
It's divided by -2.
It remains unchanged.
It's multiplied by 4.
If A is an invertible matrix and det(A) = 8, find det(A⁻¹).
8
1/8
-8
0
If two rows of a matrix are identical, what is its determinant?
1
-1
It depends on the other rows.
0
If A is a 2x2 matrix and det(A) = 0, what can be said about A?
A is invertible
A is not invertible
A is the identity matrix
A is the zero matrix
What is the rank of the matrix: [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
1
2
3
0
Solve the following system of equations using matrices: x + y = 2; 2x + 3y = 5
x = 1, y = 1
x = -1, y = 3
x = 2, y = 0
x = 0, y = 2
Given that A is a 4x4 matrix and det(A) = 6, find det(Aᵀ).
6
-6
0
1/6
If you add 5 times the first row to the second row of a matrix, what happens to its determinant?
It's multiplied by 5.
It changes sign.
It remains unchanged.
It becomes 0.
If A is a 2x2 matrix and det(A) = -3, what is det(2A)?
-3
-6
-12
-24
What is the size of the matrix product AB, given that A is a 3x4 matrix and B is a 4x2 matrix?
3x4
4x2
3x2
The product is undefined.
If A is an invertible matrix, then det(A⁻¹) is equal to:
0
1
1/det(A)
-det(A)
The trace of a matrix is the sum of its:
all entries
diagonal entries
eigenvalues
entries in the first row
If a square matrix has a column of all zeros, what is its determinant?
1
-1
It depends on other columns.
0
If A is an n x n matrix and rank(A) = n, then A is:
singular
invertible
the zero matrix
the identity matrix
If the determinant of a coefficient matrix is zero, what can you conclude about the corresponding system of linear equations?
It has a unique solution.
It has no solution.
It has infinitely many solutions.
It has no solution or infinitely many solutions.
If two columns of a matrix are interchanged, its determinant is:
unchanged
multiplied by 2
multiplied by -1
divided by 2
What is the dimension of the vector space of all 2x2 matrices?
1
2
4
Infinite
Find the determinant of the matrix: [[3, 0, 0], [0, -2, 0], [0, 0, 5]]
0
1
-30
30
If A is a 3x3 matrix with det(A) = -4, what is det(-A)?
4
-4
1/4
0
If the system of equations Ax = 0 has only the trivial solution (x = 0), then A is:
singular
non-singular
the zero matrix
the identity matrix
A system of linear equations is underdetermined if:
It has more equations than unknowns.
It has more unknowns than equations.
It has a unique solution.
The determinant of the coefficient matrix is non-zero.
What operation on a matrix does NOT change its determinant?
Swapping two rows
Multiplying a row by a scalar
Adding a multiple of one row to another row
Taking the transpose
If A is an idempotent matrix (A² = A), then what are the possible eigenvalues of A?
Only 1
Only 0
0 or 1
Any real number
A square matrix A is called nilpotent if:
A² = A
Aᵀ = A
Aᵏ = 0 for some positive integer k
det(A) = 0
Find the rank of the matrix: [[1, 2, 3], [0, 0, 0], [4, 5, 6]]
0
1
2
3
If the reduced row echelon form of the augmented matrix [A|b] has a row of the form [0 0 0 ... 0 | 1], what can be said about the system?
It has infinitely many solutions.
It has a unique solution.
It has no solution.
It is a homogeneous system.
What is the condition for Cramer's rule to be applicable to a system of linear equations?
The determinant of the coefficient matrix must be zero.
The system must have more variables than equations.
The system must be homogeneous.
The determinant of the coefficient matrix must be nonzero.
If a square matrix A satisfies Aᵀ = -A, then A is called:
symmetric
orthogonal
skew-symmetric
idempotent
Let u = (1, 2, -1) and v = (3, 0, 2). Compute 3u + 2v.
(9, 6, 1)
(9, 6, -1)
(3, 6, 1)
(3, 6, -1)
Calculate the dot product of u = (2, -1, 4) and v = (1, 3, -2).
3
-3
5
-5
Find the length (magnitude) of the vector v = (-4, 3).
1
5
7
25
Normalize the vector u = (2, -1).
(2/√5, -1/√5)
(2/5, -1/5)
(2, -1)
(√5/2, -√5)
Find the determinant of the matrix A = [[1, 2], [3, 4]].
2
-2
10
11
Calculate the determinant of the matrix B = [[2, 0, 1], [3, -1, 2], [1, 1, 0]].
7
-7
11
-11
If A is a 3x3 matrix and det(A) = 4, what is det(2A)?
4
8
32
64
If A is a 2x2 matrix and det(A) = 5, what is det(Aᵀ)?
5
-5
1/5
0
If you multiply a single row of a matrix by 3, its determinant is multiplied by:
1/3
3
9
27
If two columns of a matrix are identical, what is its determinant?
1
-1
2
0
Solve for x and y using matrices: x + y = 3; x - y = 1
x = 1, y = 2
x = 2, y = 1
x = 3, y = 0
x = 0, y = 3
What is the inverse of the matrix: [[2, 0], [0, -1]]?
[[1/2, 0], [0, -1]]
[[2, 0], [0, -1]]
[[-1, 0], [0, 2]]
The inverse doesn't exist.
If A and B are invertible nxn matrices, then (AB)⁻¹ is:
A⁻¹B⁻¹
B⁻¹A⁻¹
A⁻¹ + B⁻¹
AB
What is the trace of the matrix C = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
15
18
21
27
If A is a 3x3 matrix and det(A) = -2, what is det(-A)?
2
-2
0
1/2
If the vectors u = (1, 2) and v = (3, 6) are linearly dependent, then one is a scalar multiple of the other. What is this scalar multiple?
1/3
3
1/2
2
