Linear Algebra — MCQs & Practice Questions

72 multiple-choice questions, organised by chapter, with answers you can check as you go. Free, no sign-up.

New to a topic? Read the Linear Algebra notes first, then come back to practise.

01

Vectors

Multiple choice · 6

Q1What is [3, -2] + [1, 5]?

  1. A[4, 3]
  2. B[2, 7]
  3. C[4, -7]
  4. D[3, 3]
Show answer

Correct answer: A — [4, 3]

Add component by component: 3+1 = 4 and −2+5 = 3, giving [4, 3].

Q2A scalar of −1 applied to a vector does what?

  1. ADoubles its length
  2. BLeaves it unchanged
  3. CReverses its direction, same length
  4. DMakes it perpendicular
Show answer

Correct answer: C — Reverses its direction, same length

Multiplying by −1 flips the arrow to point the opposite way while keeping the same length.

Q3The magnitude of [6, 8] is:

  1. A14
  2. B10
  3. C48
  4. D√14
Show answer

Correct answer: B — 10

√(6² + 8²) = √(36 + 64) = √100 = 10 — the classic 6-8-10 right triangle.

Q4Two vectors have a dot product of 0. They are:

  1. AParallel
  2. BEqual
  3. CPerpendicular
  4. DBoth zero vectors
Show answer

Correct answer: C — Perpendicular

A zero dot product means the angle between them is 90° — they are perpendicular (orthogonal).

Q5Which expression is a linear combination of u and v?

  1. Au · v
  2. B|u| + |v|
  3. C3u − 2v
  4. Dthe angle between u and v
Show answer

Correct answer: C — 3u − 2v

A linear combination is scaled vectors added together, like 3u − 2v. The others produce numbers, not a combined vector.

Q6What is [2, 1] · [3, -4]?

  1. A2
  2. B10
  3. C-2
  4. D5
Show answer

Correct answer: A — 2

(2)(3) + (1)(−4) = 6 − 4 = 2. Multiply matching components and add.

02

Linear Combinations, Span & Independence

Multiple choice · 6

Q1The span of the single vector [2, 1] is:

  1. AThe whole plane
  2. BA line through the origin
  3. CJust the point [2, 1]
  4. DThe two axes
Show answer

Correct answer: B — A line through the origin

Every scalar multiple of [2, 1] lies on one line through the origin, so a single non-zero vector spans a line.

Q2Which pair is linearly DEPENDENT?

  1. A[1, 0] and [0, 1]
  2. B[1, 2] and [2, 4]
  3. C[1, 1] and [1, -1]
  4. D[3, 0] and [0, 5]
Show answer

Correct answer: B — [1, 2] and [2, 4]

[2, 4] = 2·[1, 2], so one is a multiple of the other — dependent. The others are independent pairs.

Q3A set of vectors is linearly independent exactly when:

  1. AThey all have length 1
  2. BThe only combination giving the zero vector is all-zero coefficients
  3. CThey are perpendicular
  4. DThere are at least three of them
Show answer

Correct answer: B — The only combination giving the zero vector is all-zero coefficients

That is the definition: no non-trivial combination collapses to zero, so none is redundant.

Q4A basis for a space is a set that is:

  1. ASpanning but dependent
  2. BIndependent but too small to span
  3. CBoth independent and spanning
  4. DAny set of perpendicular vectors
Show answer

Correct answer: C — Both independent and spanning

A basis must reach everything (span) with nothing wasted (independent).

Q5Three vectors in 2D must be:

  1. AIndependent
  2. BA basis
  3. CLinearly dependent
  4. DPerpendicular
Show answer

Correct answer: C — Linearly dependent

The plane has dimension 2, so any three vectors in it are necessarily dependent — one is a combination of the others.

Q6The columns of a 3×3 matrix are independent. Its determinant is:

  1. AZero
  2. BNon-zero
  3. CAlways 1
  4. DNegative
Show answer

Correct answer: B — Non-zero

Independent columns ⇔ non-zero determinant ⇔ the matrix is invertible.

03

Matrices

Multiple choice · 6

Q1What shape results from multiplying a (2×3) matrix by a (3×4) matrix?

  1. A2×4
  2. B3×3
  3. C2×3
  4. DIt is not defined
Show answer

Correct answer: A — 2×4

The inner 3s match and cancel; the outer dimensions 2 and 4 give a 2×4 result.

Q2For A = [[2, 0], [1, 3]] and v = [1, 2], what is A·v?

  1. A[2, 7]
  2. B[2, 6]
  3. C[3, 5]
  4. D[4, 2]
Show answer

Correct answer: A — [2, 7]

Top row [2,0]·[1,2] = 2; bottom row [1,3]·[1,2] = 1 + 6 = 7. So [2, 7].

Q3Which statement about matrix multiplication is TRUE?

  1. AAB always equals BA
  2. BAB and BA are usually different
  3. COnly 2×2 matrices multiply
  4. DYou add matching entries
Show answer

Correct answer: B — AB and BA are usually different

Matrix multiplication is not commutative; AB and BA generally differ, and often only one is a legal shape.

Q4What does the identity matrix do when it multiplies a vector v?

  1. ADoubles v
  2. BReturns v unchanged
  3. CReturns the zero vector
  4. DTransposes v
Show answer

Correct answer: B — Returns v unchanged

The identity is the do-nothing matrix: I·v = v, just as 1 × a number leaves it unchanged.

Q5Reading A·[x, y] as a combination of A’s columns c₁ and c₂ gives:

  1. Ax·c₁ + y·c₂
  2. Bc₁ · c₂
  3. Cx·c₂ + y·c₁
  4. D(x + y)(c₁ + c₂)
Show answer

Correct answer: A — x·c₁ + y·c₂

The input’s components are the weights: A·[x,y] = x·c₁ + y·c₂, a linear combination of the columns.

Q6The transpose of a matrix is formed by:

  1. ANegating every entry
  2. BFlipping it across its main diagonal
  3. CSwapping its two largest entries
  4. DMultiplying by the identity
Show answer

Correct answer: B — Flipping it across its main diagonal

Transposing reflects the matrix across the diagonal, turning row i into column i.

04

Linear Transformations

Multiple choice · 6

Q1The columns of a 2×2 matrix tell you:

  1. AIts determinant
  2. BWhere î and ĵ land after the transformation
  3. CIts inverse
  4. DThe angle of rotation only
Show answer

Correct answer: B — Where î and ĵ land after the transformation

Column 1 is the image of î and column 2 is the image of ĵ; those landing spots define the whole transformation.

Q2A linear transformation must keep which point fixed?

  1. AThe point (1,1)
  2. BThe origin
  3. CThe top-right corner
  4. DNo point is fixed
Show answer

Correct answer: B — The origin

Linearity fixes the origin; grid lines stay straight and evenly spaced around it.

Q3The matrix [[2, 0], [0, 2]] does what to space?

  1. ARotates it 90°
  2. BReflects it
  3. CScales everything by 2 away from the origin
  4. DCollapses it to a line
Show answer

Correct answer: C — Scales everything by 2 away from the origin

î → [2,0] and ĵ → [0,2], so every vector doubles its distance from the origin — a uniform scaling.

Q4The product BA applied to a vector v means:

  1. ADo B first, then A
  2. BDo A first, then B
  3. CAdd the two transformations
  4. DThe same as AB
Show answer

Correct answer: B — Do A first, then B

Composition reads right to left: BA(v) = B(A(v)), so A acts first.

Q5Where does [[0, -1], [1, 0]] send the vector [1, 0]?

  1. A[0, 1]
  2. B[-1, 0]
  3. C[1, 0]
  4. D[0, -1]
Show answer

Correct answer: A — [0, 1]

[1,0] is î, which lands on the first column of the matrix: [0, 1] — a 90° anticlockwise turn.

Q6If both columns of a matrix point along the same line, the transformation:

  1. AIs a rotation
  2. BPreserves all areas
  3. CSquashes the plane onto a line
  4. DIs the identity
Show answer

Correct answer: C — Squashes the plane onto a line

When î and ĵ land on one line, the whole plane is flattened onto that line — the matrix is singular.

05

Systems of Linear Equations

Multiple choice · 6

Q1How many solutions can a system of linear equations have?

  1. AOnly one, ever
  2. BOne, none, or infinitely many
  3. CAny whole number
  4. DAt most two
Show answer

Correct answer: B — One, none, or infinitely many

Linearity forces exactly one of three outcomes: a unique solution, no solution, or infinitely many.

Q2Two lines from a 2-equation system are parallel and distinct. The system has:

  1. AOne solution
  2. BNo solution
  3. CInfinitely many solutions
  4. DExactly two solutions
Show answer

Correct answer: B — No solution

Parallel distinct lines never meet, so there is no point satisfying both equations.

Q3Which is NOT a legal row operation in Gaussian elimination?

  1. ASwapping two rows
  2. BMultiplying a row by 0
  3. CAdding a multiple of one row to another
  4. DMultiplying a row by 3
Show answer

Correct answer: B — Multiplying a row by 0

Multiplying a row by zero destroys information; scaling must be by a non-zero number.

Q4A column with no pivot after elimination indicates:

  1. Aa unique solution
  2. Ba free variable, so infinitely many solutions
  3. Cno solution is possible
  4. Dan arithmetic mistake
Show answer

Correct answer: B — a free variable, so infinitely many solutions

A pivot-free column is a free variable, which lets the system have infinitely many solutions.

Q5Eliminate x from x + y = 3, 2x + 3y = 8. What does row 2 become?

  1. Ay = 2
  2. By = 5
  3. Cx = 1
  4. D0 = 1
Show answer

Correct answer: A — y = 2

R2 − 2·R1: (2−2)x + (3−2)y = 8 − 6 → y = 2. Then x = 1.

Q6A square system with det(A) ≠ 0 has:

  1. ANo solution
  2. BExactly one solution
  3. CInfinitely many solutions
  4. DA free variable
Show answer

Correct answer: B — Exactly one solution

Non-zero determinant means A is invertible, so Ax = b has the single solution x = A⁻¹b.

06

Inverse Matrices

Multiple choice · 6

Q1A matrix A is invertible if and only if:

  1. AIt is square
  2. BIts determinant is non-zero
  3. CAll its entries are positive
  4. DIt is symmetric
Show answer

Correct answer: B — Its determinant is non-zero

Square is necessary but not enough; the deciding condition is det(A) ≠ 0.

Q2What is the determinant of [[4, 3], [2, 2]] (needed for its inverse)?

  1. A2
  2. B14
  3. C8
  4. D-2
Show answer

Correct answer: A — 2

ad − bc = (4)(2) − (3)(2) = 8 − 6 = 2, which is non-zero, so the inverse exists.

Q3A singular matrix is one that:

  1. AHas an inverse
  2. BHas determinant 0 and no inverse
  3. CIs always the identity
  4. DHas only one row
Show answer

Correct answer: B — Has determinant 0 and no inverse

Singular means det = 0: space was collapsed, so the transformation cannot be undone.

Q4The 2×2 inverse formula divides by:

  1. Aa + d
  2. Bthe determinant ad − bc
  3. Cthe trace
  4. Dbc − ad only
Show answer

Correct answer: B — the determinant ad − bc

A⁻¹ = 1/(ad − bc) · [[d, −b], [−c, a]]; dividing by zero is exactly why singular matrices fail.

Q5What is (AB)⁻¹?

  1. AA⁻¹B⁻¹
  2. BB⁻¹A⁻¹
  3. CA⁻¹ + B⁻¹
  4. D(BA)⁻¹
Show answer

Correct answer: B — B⁻¹A⁻¹

To undo "do A, then B" you undo B first: (AB)⁻¹ = B⁻¹A⁻¹.

Q6The best way to solve a single system Ax = b is usually:

  1. ACompute A⁻¹ then multiply
  2. BGaussian elimination
  3. CGuess and check
  4. DFind the eigenvalues
Show answer

Correct answer: B — Gaussian elimination

Elimination is faster and numerically safer than forming the inverse just to solve once.

07

Determinants

Multiple choice · 6

Q1What is det[[4, 2], [3, 1]]?

  1. A-2
  2. B10
  3. C2
  4. D7
Show answer

Correct answer: A — -2

ad − bc = (4)(1) − (2)(3) = 4 − 6 = −2. The negative sign means orientation is flipped.

Q2A matrix has determinant 0. What follows?

  1. AIt doubles every area
  2. BIt is singular and has no inverse
  3. CIt is a rotation
  4. DIt is the identity
Show answer

Correct answer: B — It is singular and has no inverse

Determinant 0 means space was flattened; the matrix is singular and cannot be inverted.

Q3A determinant of −5 means the transformation:

  1. AIs impossible
  2. BScales area by 5 and flips orientation
  3. CShrinks everything to a point
  4. DPreserves area
Show answer

Correct answer: B — Scales area by 5 and flips orientation

The magnitude 5 is the area scaling; the negative sign records that space was flipped over.

Q4If det(A) = 2 and det(B) = 3, what is det(AB)?

  1. A5
  2. B6
  3. C1
  4. DCannot be determined
Show answer

Correct answer: B — 6

The product rule says det(AB) = det(A)·det(B), because applying B then A scales area by both factors in turn: 2 × 3 = 6. This is why the rule matters — you never have to multiply the matrices out to find the determinant of a composition.

Q5Which matrix preserves area (determinant 1)?

  1. A[[2, 0], [0, 2]]
  2. B[[1, 1], [0, 1]]
  3. C[[0, 0], [0, 0]]
  4. D[[3, 0], [0, 3]]
Show answer

Correct answer: B — [[1, 1], [0, 1]]

The shear [[1,1],[0,1]] has det = (1)(1) − (1)(0) = 1 — it slides space without changing area.

Q6The determinant equals zero exactly when the columns of the matrix are:

  1. APerpendicular
  2. BUnit length
  3. COn the same line (linearly dependent)
  4. DBoth positive
Show answer

Correct answer: C — On the same line (linearly dependent)

If the columns lie on one line they span no area, the parallelogram is flat, and the determinant is 0.

08

Vector Spaces & the Four Subspaces

Multiple choice · 6

Q1Which of these is a subspace of the plane?

  1. AA line through the origin
  2. BA line that misses the origin
  3. CA single non-zero point
  4. DA circle
Show answer

Correct answer: A — A line through the origin

A subspace must be closed under scaling and contain 0, so only a line through the origin qualifies.

Q2The column space of A is:

  1. AAll x with Ax = 0
  2. BThe span of the columns of A
  3. CThe determinant of A
  4. DThe largest entry of A
Show answer

Correct answer: B — The span of the columns of A

The column space is the span of the columns — every b reachable as Ax.

Q3Ax = b has a solution precisely when:

  1. AA is square
  2. Bb is in the column space of A
  3. Cdet(A) = 0
  4. Db is the zero vector
Show answer

Correct answer: B — b is in the column space of A

Ax ranges over the column space, so b must lie in it for a solution to exist.

Q4A non-trivial nullspace (more than just the zero vector) means A is:

  1. AInvertible
  2. BNot invertible
  3. CSymmetric
  4. DThe identity
Show answer

Correct answer: B — Not invertible

Vectors other than 0 mapping to 0 mean dependent columns, so A cannot be inverted.

Q5A 3-column matrix has rank 2. Its nullity is:

  1. A0
  2. B1
  3. C2
  4. D3
Show answer

Correct answer: B — 1

rank + nullity = number of columns: 2 + nullity = 3, so nullity = 1.

Q6The rank of a matrix equals:

  1. AIts number of rows
  2. BIts number of pivots
  3. CIts determinant
  4. DThe sum of its entries
Show answer

Correct answer: B — Its number of pivots

Rank is the number of pivots — the count of independent columns.

09

Orthogonality, Projections & Least Squares

Multiple choice · 6

Q1Two vectors are orthogonal exactly when:

  1. AThey have equal length
  2. BTheir dot product is zero
  3. CThey point the same way
  4. DTheir sum is zero
Show answer

Correct answer: B — Their dot product is zero

A zero dot product is the definition of a right angle between vectors.

Q2The projection of b onto the line through a is:

  1. A(a·b / a·a)·a
  2. Ba·b
  3. Ca + b
  4. D|a||b|
Show answer

Correct answer: A — (a·b / a·a)·a

proj = (a·b / a·a)·a — the multiple of a closest to b.

Q3Least squares is used when Ax = b:

  1. AHas exactly one solution
  2. BHas no exact solution
  3. CIs already diagonal
  4. DHas a zero right-hand side
Show answer

Correct answer: B — Has no exact solution

When b lies outside the column space there is no exact solution, so we find the closest x̂ instead.

Q4The normal equations for the least-squares solution are:

  1. AAx = b
  2. BAᵀA x̂ = Aᵀb
  3. CA⁻¹x = b
  4. Ddet(A) = 0
Show answer

Correct answer: B — AᵀA x̂ = Aᵀb

Setting the error orthogonal to the columns gives AᵀA x̂ = Aᵀb.

Q5In a projection, the error vector (b − proj) is always:

  1. AParallel to the line
  2. BPerpendicular to the subspace
  3. CLonger than b
  4. DThe zero vector
Show answer

Correct answer: B — Perpendicular to the subspace

The closest point leaves an error orthogonal to the subspace — the whole basis of least squares.

Q6Gram–Schmidt produces:

  1. AThe determinant
  2. BAn orthonormal basis
  3. CThe eigenvalues
  4. DThe inverse
Show answer

Correct answer: B — An orthonormal basis

It converts any basis into a mutually perpendicular, unit-length (orthonormal) basis.

10

Eigenvalues & Eigenvectors

Multiple choice · 6

Q1An eigenvector of A is a vector that, under A,

  1. ABecomes the zero vector
  2. BKeeps its direction and is only scaled
  3. CRotates by 90°
  4. DDoubles in every case
Show answer

Correct answer: B — Keeps its direction and is only scaled

By definition A v = λ v: the direction is preserved and the vector is scaled by the eigenvalue λ.

Q2The eigenvalues of A are found by solving:

  1. AA v = 0
  2. Bdet(A − λI) = 0
  3. Cdet(A) = 1
  4. DA = P D P⁻¹
Show answer

Correct answer: B — det(A − λI) = 0

The characteristic equation det(A − λI) = 0 yields the eigenvalues.

Q3Once you have an eigenvalue λ, its eigenvectors are:

  1. AThe columns of A
  2. BThe nullspace of (A − λI)
  3. CThe rows of A⁻¹
  4. DAlways [1, 0]
Show answer

Correct answer: B — The nullspace of (A − λI)

Substitute λ and solve (A − λI)v = 0; the solutions are the eigenvectors.

Q4For A = [[2, 1], [1, 2]], the eigenvalues are:

  1. A2 and 2
  2. B1 and 3
  3. C0 and 4
  4. D-1 and 3
Show answer

Correct answer: B — 1 and 3

(2−λ)² − 1 = (λ−1)(λ−3) = 0, so λ = 1 and λ = 3 (they sum to the trace 4 and multiply to det 3).

Q5Diagonalizing A = P D P⁻¹ makes which computation easy?

  1. AAdding matrices
  2. BHigh powers Aⁿ
  3. CTransposing A
  4. DCounting the rows
Show answer

Correct answer: B — High powers Aⁿ

Aⁿ = P Dⁿ P⁻¹, and powers of a diagonal matrix are found entry by entry.

Q6The eigenvalues of a matrix always multiply to give:

  1. AThe trace
  2. BThe determinant
  3. CZero
  4. DThe rank
Show answer

Correct answer: B — The determinant

The product of the eigenvalues equals the determinant; their sum equals the trace.

11

The Singular Value Decomposition

Multiple choice · 6

Q1The singular value decomposition writes a matrix as:

  1. AA = P D P⁻¹
  2. BA = U Σ Vᵀ
  3. CA = LU
  4. DA = QR
Show answer

Correct answer: B — A = U Σ Vᵀ

A = U Σ Vᵀ: rotation, stretch, rotation — valid for every matrix of any shape.

Q2Which matrices have an SVD?

  1. AOnly symmetric ones
  2. BOnly invertible square ones
  3. CEvery matrix, of any shape
  4. DOnly 2×2 matrices
Show answer

Correct answer: C — Every matrix, of any shape

Unlike eigen-decomposition, the SVD exists for every matrix without exception.

Q3The number of non-zero singular values of a matrix equals its:

  1. ADeterminant
  2. BRank
  3. CTrace
  4. DNumber of rows
Show answer

Correct answer: B — Rank

Non-zero singular values count the independent directions the matrix truly acts on — its rank.

Q4Singular values are the square roots of the eigenvalues of:

  1. AA
  2. BAᵀA
  3. CA⁻¹
  4. DA + Aᵀ
Show answer

Correct answer: B — AᵀA

AᵀA is symmetric with non-negative eigenvalues; their square roots are the singular values.

Q5Keeping only the largest singular values gives you:

  1. AThe determinant
  2. BThe best low-rank approximation (compression)
  3. CThe inverse
  4. DThe transpose
Show answer

Correct answer: B — The best low-rank approximation (compression)

Truncating to the top singular values is the best approximation of that rank — the basis of compression, denoising and PCA.

Q6Geometrically, the SVD shows every matrix is at most a:

  1. ASingle rotation
  2. BRotation, stretch along perpendicular axes, then rotation
  3. CReflection only
  4. DPure translation
Show answer

Correct answer: B — Rotation, stretch along perpendicular axes, then rotation

Vᵀ rotates, Σ stretches along perpendicular axes, U rotates again — that is all any matrix does.

12

Applications of Linear Algebra

Multiple choice · 6

Q1In computer graphics, rotating and scaling a shape is done by:

  1. AAdding vectors
  2. BMultiplying coordinates by a matrix
  3. CTaking a determinant
  4. DSorting the points
Show answer

Correct answer: B — Multiplying coordinates by a matrix

Rotations and scalings are linear transformations — matrices multiplying the coordinate vectors.

Q2Homogeneous coordinates are introduced so that ___ becomes a matrix operation.

  1. ARotation
  2. BTranslation (sliding)
  3. CScaling
  4. DReflection
Show answer

Correct answer: B — Translation (sliding)

Translation is not linear by itself; an extra coordinate lets it be written as one matrix multiplication.

Q3The steady state of a Markov chain is:

  1. AThe determinant of the matrix
  2. BThe eigenvector for eigenvalue 1
  3. CThe largest entry
  4. DAlways the zero vector
Show answer

Correct answer: B — The eigenvector for eigenvalue 1

Over time the system settles into the λ = 1 eigenvector — its steady-state distribution.

Q4Google's original PageRank computes:

  1. AA determinant of the web
  2. BThe dominant eigenvector of the link matrix
  3. CThe inverse of the web matrix
  4. DThe trace of the network
Show answer

Correct answer: B — The dominant eigenvector of the link matrix

PageRank is the steady-state (dominant) eigenvector of the web's Markov link matrix.

Q5Principal Component Analysis reduces data dimensions using:

  1. ARandom sampling
  2. BEigenvectors or the SVD of the data
  3. CThe identity matrix
  4. DRow swaps
Show answer

Correct answer: B — Eigenvectors or the SVD of the data

PCA keeps the top eigen/singular directions — those carrying the most variation in the data.

Q6A neural network is, at its core, mostly:

  1. ASorting algorithms
  2. BStacked matrix multiplications
  3. CDeterminant calculations
  4. DGraph colouring
Show answer

Correct answer: B — Stacked matrix multiplications

Layers are matrix multiplications with simple functions between them; training adjusts the matrix entries.

These questions come from the Linear Algebra lessons — each topic has its own notes, worked examples and an interactive diagram.