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Practice deliberately

Work through Linear Algebra unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Linear Algebra coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Linear Algebra syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use linear algebra practice questions with answers.

01

Systems and elimination

Represent the constraints symbolically, choose an equivalence-preserving solution method, and verify the result in every original relationship.

Goal: recognise a systems and elimination problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Matrix algebra

Track dimensions and geometric meaning while computing, then verify the result by multiplication, substitution, or a basis-vector test.

Goal: recognise a matrix algebra problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Determinants

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a determinants problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Vector spaces

Track dimensions and geometric meaning while computing, then verify the result by multiplication, substitution, or a basis-vector test.

Goal: recognise a vector spaces problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Linear transformations

Connect formulas, tables, and graphs; identify domain and parameters; then interpret how changing an input or coefficient changes the output.

Goal: recognise a linear transformations problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Eigenvalues

Track dimensions and geometric meaning while computing, then verify the result by multiplication, substitution, or a basis-vector test.

Goal: recognise a eigenvalues problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Orthogonality

Track dimensions and geometric meaning while computing, then verify the result by multiplication, substitution, or a basis-vector test.

Goal: recognise a orthogonality problem from its wording, carry out the governing method, and check that the result is reasonable.

Linear Algebra questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Solve x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2.

Linear Algebra · Systems and elimination

  1. x = 1, y = 2, z = 3
  2. x = 3, y = 2, z = 1
  3. x = 2, y = 1, z = 3
  4. x = 1, y = 3, z = 2
Show the worked answer
  1. Subtract twice the first equation from the second: -3y - z = -9.
  2. Subtract the first equation from the third: y - 2z = -4.
  3. From the second result, z = 9 - 3y. Substituting gives y - 18 + 6y = -4, so 7y = 14 and y = 2.
  4. Then z = 9 - 6 = 3 and x = 6 - 2 - 3 = 1.

Answer: x = 1, y = 2, z = 3

Question 2: Multiply A = [[1, 2], [3, 4]] by B = [[0, 1], [-1, 2]].

Linear Algebra · Matrix algebra

  1. [[-2, 5], [-4, 11]]
  2. [[-2, 5], [-4, 10]]
  3. [[0, 2], [-3, 8]]
  4. [[2, 5], [4, 11]]
Show the worked answer
  1. Each entry is a row of A dotted with a column of B.
  2. Row 1 with column 1: 1(0) + 2(-1) = -2. Row 1 with column 2: 1(1) + 2(2) = 5.
  3. Row 2 with column 1: 3(0) + 4(-1) = -4.
  4. Row 2 with column 2: 3(1) + 4(2) = 11. So AB = [[-2, 5], [-4, 11]].

Answer: [[-2, 5], [-4, 11]]

Question 3: Find the determinant of [[2, 0, 1], [3, -1, 2], [1, 4, 0]].

Linear Algebra · Determinants

  1. -3
  2. 3
  3. -11
  4. 13
Show the worked answer
  1. Expand along the first row, where the middle entry is 0 and can be skipped.
  2. The 2 term: 2 x [(-1)(0) - (2)(4)] = 2(-8) = -16.
  3. The 1 term in position three: +1 x [(3)(4) - (-1)(1)] = 12 + 1 = 13.
  4. -16 + 13 = -3.

Answer: -3

Question 4: A 3 by 4 matrix has rank 2. What is the dimension of its null space?

Linear Algebra · Vector spaces

  1. 2
  2. 1
  3. 3
  4. 4
Show the worked answer
  1. The rank-nullity theorem says rank plus nullity equals the number of columns.
  2. This matrix has 4 columns.
  3. 4 - 2 = 2.
  4. So the null space has dimension 2.

Answer: 2

Question 5: A linear map T sends (1, 0) to (2, 3) and (0, 1) to (-1, 4). What is T(3, 2)?

Linear Algebra · Linear transformations

  1. (4, 17)
  2. (5, 14)
  3. (4, 14)
  4. (8, 5)
Show the worked answer
  1. A linear map is decided by what it does to the basis vectors.
  2. (3, 2) = 3(1, 0) + 2(0, 1), so T(3, 2) = 3(2, 3) + 2(-1, 4).
  3. 3(2, 3) = (6, 9) and 2(-1, 4) = (-2, 8).
  4. Add: (6 - 2, 9 + 8) = (4, 17).

Answer: (4, 17)

Question 6: Find the eigenvalues of [[4, 1], [2, 3]].

Linear Algebra · Eigenvalues

  1. 5 and 2
  2. 4 and 3
  3. 5 and -2
  4. 7 and 10
Show the worked answer
  1. The characteristic equation is lambda^2 - (trace) lambda + (determinant) = 0.
  2. The trace is 4 + 3 = 7 and the determinant is 4(3) - 1(2) = 10.
  3. So lambda^2 - 7 lambda + 10 = 0, which factors as (lambda - 5)(lambda - 2).
  4. The eigenvalues are 5 and 2.

Answer: 5 and 2

Question 7: What is the projection of the vector (4, 3) onto the vector (1, 1)?

Linear Algebra · Orthogonality

  1. (3.5, 3.5)
  2. (7, 7)
  3. (2, 2)
  4. (4, 4)
Show the worked answer
  1. The projection is (u dot v) / (v dot v) times v.
  2. u dot v = 4(1) + 3(1) = 7.
  3. v dot v = 1 + 1 = 2, so the scale factor is 7 / 2 = 3.5.
  4. 3.5 x (1, 1) = (3.5, 3.5).

Answer: (3.5, 3.5)

Question 8: What is the inverse of [[3, 5], [1, 2]]?

Linear Algebra · Matrix algebra

  1. [[2, -5], [-1, 3]]
  2. [[2, 5], [1, 3]]
  3. [[3, -5], [-1, 2]]
  4. [[-2, 5], [1, -3]]
Show the worked answer
  1. For a 2 by 2 matrix, swap the diagonal entries, negate the other two, and divide by the determinant.
  2. The determinant is 3(2) - 5(1) = 1.
  3. Dividing by 1 changes nothing.
  4. The inverse is [[2, -5], [-1, 3]]. Check: multiplying gives the identity matrix.

Answer: [[2, -5], [-1, 3]]

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Linear Algebra questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Systems and eliminationSolve x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2.x = 1, y = 2, z = 3
2Matrix algebraMultiply A = [[1, 2], [3, 4]] by B = [[0, 1], [-1, 2]].[[-2, 5], [-4, 11]]
3DeterminantsFind the determinant of [[2, 0, 1], [3, -1, 2], [1, 4, 0]].-3
4Vector spacesA 3 by 4 matrix has rank 2. What is the dimension of its null space?2
5Linear transformationsA linear map T sends (1, 0) to (2, 3) and (0, 1) to (-1, 4). What is T(3, 2)?(4, 17)
6EigenvaluesFind the eigenvalues of [[4, 1], [2, 3]].5 and 2
7OrthogonalityWhat is the projection of the vector (4, 3) onto the vector (1, 1)?(3.5, 3.5)
8Matrix algebraWhat is the inverse of [[3, 5], [1, 2]]?[[2, -5], [-1, 3]]

How the generated Linear Algebra sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Linear Algebra practice page

What does this page cover?

It covers all 7 Linear Algebra units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.