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Question 1
Linear AlgebraFoundation

Linear Algebra practice question 1

Linear Algebra practice topics

Question sets mix direct skills with unfamiliar applications so method selection becomes part of the practice.

01

Matrix operations and elimination

Add and multiply matrices in the correct order, and reduce a system to row echelon form to solve it.

02

Vector spaces and linear maps

Test vectors for linear independence, find the dimension of a span, and apply rank-nullity to a linear map.

03

Determinants and rank

Compute determinants for 2x2 and 3x3 matrices, and count independent rows to decide rank and invertibility.

04

Eigenvalues and orthogonality

Find eigenvalues from the trace and determinant, and use dot products to test for orthogonality and build projections.

Linear Algebra practice questions with worked answers

These 8 questions are printed in full on this page, with every step of the arithmetic written out. Cover the options, solve the question on paper first, and only then open the worked answer to compare your method with the one shown.

Question 1: Multiply the matrices [[2, 1], [0, 3]] and [[1, 4], [2, 5]], in that order.

Foundation level · matrix operations and elimination

  1. [[4, 13], [6, 15]]
  2. [[2, 4], [0, 15]]
  3. [[4, 6], [13, 15]]
  4. [[3, 5], [2, 8]]
Show the worked answer
  1. Each entry is a row of the first matrix dotted with a column of the second.
  2. Top-left: 2(1) + 1(2) = 4. Top-right: 2(4) + 1(5) = 8 + 5 = 13.
  3. Bottom-left: 0(1) + 3(2) = 6. Bottom-right: 0(4) + 3(5) = 15.
  4. So the product is [[4, 13], [6, 15]]. Matrix order matters, so this is not the same as swapping them.

Answer: [[4, 13], [6, 15]]

Question 2: Use elimination to solve x + 2y = 7 and 3x - y = 7.

Exam level · matrix operations and elimination

  1. x = 3, y = 2
  2. x = 2, y = 3
  3. x = 1, y = 3
  4. x = 5, y = 1
Show the worked answer
  1. Multiply the second equation by 2 so the y terms cancel: 6x - 2y = 14.
  2. Add it to the first equation: (x + 6x) + (2y - 2y) = 7 + 14, so 7x = 21.
  3. Divide by 7: x = 3.
  4. Substitute back: 3 + 2y = 7, so 2y = 4 and y = 2.

Answer: x = 3, y = 2

Question 3: Find the determinant of the matrix [[4, 3], [2, 5]].

Foundation level · determinants and rank

  1. 14
  2. 26
  3. 20
  4. -14
Show the worked answer
  1. For a 2 by 2 matrix [[a, b], [c, d]] the determinant is ad - bc.
  2. Here ad = 4 x 5 = 20.
  3. And bc = 3 x 2 = 6.
  4. 20 - 6 = 14, so the determinant is 14. It is not zero, so the matrix is invertible.

Answer: 14

Question 4: What is the rank of the matrix [[1, 2, 3], [2, 4, 6], [1, 1, 1]]?

Exam level · determinants and rank

  1. 2
  2. 3
  3. 1
  4. 0
Show the worked answer
  1. Rank counts the independent rows, so look for rows that are copies of one another.
  2. Row 2 is exactly 2 times Row 1: 2(1, 2, 3) = (2, 4, 6), so it adds nothing new.
  3. Row 1 and Row 3 are not multiples of each other, since 1(1, 2, 3) is not (1, 1, 1).
  4. That leaves 2 independent rows, so the rank is 2.

Answer: 2

Question 5: Vectors u = (1, 2, 3) and v = (2, 4, 6) are given. What is the dimension of the space they span?

Foundation level · vector spaces and linear maps

  1. 1
  2. 2
  3. 3
  4. 0
Show the worked answer
  1. Check whether one vector is a multiple of the other.
  2. 2 x (1, 2, 3) = (2, 4, 6), which is exactly v.
  3. So v lies on the same line through the origin as u and adds no new direction.
  4. The span is that single line, so its dimension is 1.

Answer: 1

Question 6: A linear map T from R^4 to R^3 has rank 3. What is the dimension of its kernel?

Challenge level · vector spaces and linear maps

  1. 1
  2. 0
  3. 3
  4. 4
Show the worked answer
  1. The rank-nullity theorem says rank + nullity = the dimension of the input space.
  2. The input space is R^4, so rank + nullity = 4.
  3. The rank is given as 3, so nullity = 4 - 3.
  4. 4 - 3 = 1, so the kernel has dimension 1: a single line of vectors is sent to zero.

Answer: 1

Question 7: Find the eigenvalues of the matrix [[4, 1], [2, 3]].

Exam level · eigenvalues and orthogonality

  1. 5 and 2
  2. 4 and 3
  3. 7 and 10
  4. 5 and -2
Show the worked answer
  1. For a 2 by 2 matrix the eigenvalues satisfy L^2 - (trace)L + (determinant) = 0.
  2. The trace is 4 + 3 = 7, and the determinant is 4(3) - 1(2) = 12 - 2 = 10.
  3. So solve L^2 - 7L + 10 = 0, which factors as (L - 5)(L - 2) = 0.
  4. The eigenvalues are 5 and 2. Check: 5 + 2 = 7 and 5 x 2 = 10.

Answer: 5 and 2

Question 8: For what value of k are the vectors (2, -1, 4) and (3, 2, k) orthogonal?

Challenge level · eigenvalues and orthogonality

  1. k = -1
  2. k = 1
  3. k = 0
  4. k = -4
Show the worked answer
  1. Two vectors are orthogonal exactly when their dot product is zero.
  2. The dot product is 2(3) + (-1)(2) + 4k = 6 - 2 + 4k.
  3. Set it to zero: 4 + 4k = 0, so 4k = -4.
  4. k = -1. Check: 6 - 2 - 4 = 0.

Answer: k = -1

How the three Linear Algebra difficulty levels differ

The difficulty buttons above change what a question asks of you, not just the size of the numbers. Each example below is taken from the question set on this page.

Difficulty levels for Linear Algebra practice, with an example question from this page.
LevelWhat it testsExample questionTime target
FoundationOne skill at a time, with the numbers kept small enough to check in your head.Multiply the matrices [[2, 1], [0, 3]] and [[1, 4], [2, 5]], in that order.About 1 minute
ExamThe wording of a real test paper: pick the method first, then carry out two or three steps.Use elimination to solve x + 2y = 7 and 3x - y = 7.2 to 3 minutes
ChallengeTwo ideas combined, or a result you have to interpret after the calculation ends.A linear map T from R^4 to R^3 has rank 3. What is the dimension of its kernel?4 to 5 minutes

How to practice Linear Algebra effectively

Begin without notes and explain your choice before checking. For every miss, identify whether the cause was a definition, setup, calculation, interpretation, or time decision. Re-solve the question from a blank page, then return to the same skill in a mixed set tomorrow.

What your Linear Algebra answer review should show

A useful review shows more than the correct option. Compare the method with your first attempt, locate the earliest incorrect decision, and write one rule that would prevent the same error in a new Linear Algebra problem.

Move from mixed Linear Algebra questions to a complete course

This page targets flexible Linear Algebra question practice. When you need a syllabus-aligned sequence with unit selection, use Linear Algebra practice by unit and return here later for mixed retrieval.

Questions about Linear Algebra practice

When should I change the difficulty?

Move up after you can solve several questions accurately without hints and explain the method. Move down for one short set when errors show that a definition or setup is still uncertain.

How often should I practice?

Short sessions on several days usually build stronger recall than one long session. Revisit missed Linear Algebra skills the next day, then mix them with older topics later in the week.

Where can I review the lessons in order?

Use Linear Algebra practice by unit for a syllabus-aligned sequence with unit selection, practice, and a complete answer review.

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