📖 What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. We describe its size as m × n (m rows, n columns). Matrices let us represent and solve systems of equations, transform coordinates, and model real-world engineering problems compactly.

Rows × ColumnsElements / entries Square matrixIdentity matrix I

Types of matrix

Row matrix — 1 × n e.g. [372]

Column matrix — m × 1 e.g. [419]

Square matrix — m = n e.g. 2×2, 3×3

Zero matrix — all entries are 0

Identity I — 1s on diagonal, 0s elsewhere

Identity matrix

The identity matrix I is the matrix equivalent of 1 — any matrix multiplied by I gives itself: A × I = A

I₂ = [1001]

I₃ = [100010001]

➕

Matrix Addition & Subtraction

Rule: matrices must be the same size. Add or subtract the corresponding elements — same row, same column.

A + B: (A+B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ A − B: (A−B)ᵢⱼ = Aᵢⱼ − Bᵢⱼ Sizes must match!

🎮 Interactive — edit any cell, watch the result update live

Enter values in Matrix A and B above — result updates instantly.

👆Try Switch to 3×3 using the size button · Click Subtract A−B to flip the operation · Make A = B then subtract — every element becomes zero!

Worked example — 2×2 addition

A + B where A=[1234]and B=[5678]

Position (1,1):1 + 5 = 6

Position (1,2):2 + 6 = 8

Position (2,1):3 + 7 = 10

Position (2,2):4 + 8 = 12

Result:[681012] ✓

Key rules

✅ Same size → can add/subtract

❌ Different sizes → NOT possible

A + B = B + A (commutative)

(A + B) + C = A + (B + C) (associative)

💡 Scalar multiplication: 3A means multiply every element by 3

✖️

Matrix Multiplication

Each result cell C_ij = Row i of A dotted with Column j of B. The guide below shows you one cell at a time.

Rule: A (m×n) × B (n×p) → result C (m×p). Inner dimensions must match. A×B ≠ B×A in general.

MATRIX A

×

MATRIX B

=

RESULT C

Press Start Guided Tour to walk through the multiplication cell by cell.

👆Press ▶ Start Guided Tour to walk through each cell one at a time · Try 3×3 mode · Edit any cell and press Start again to see your own multiplication worked out.

Worked: 2×2

A=[3213]× B=[124−6]

C₁₁:Row1·Col1 = 3×1+2×4 = 11

C₁₂:Row1·Col2 = 3×2+2×(−6) = −6

C₂₁:Row2·Col1 = 1×1+3×4 = 13

C₂₂:Row2·Col2 = 1×2+3×(−6) = −16

C =[11−613−16] ✓

Key rules

A(m×n) × B(n×p) → C is m×p

A×B ≠ B×A (not commutative)

2×2 × 3×2? → Cols of A ≠ Rows of B → ❌ Impossible

A×(B+C) = A×B + A×C (distributive)

💡 A×B is possible only when the inner numbers match: A(m×n) × B(n×p)

🔢

Determinant

The determinant is a single number from a square matrix. For 2×2: det = ad − bc. For 3×3 expand along row 1. If det = 0 the matrix has no inverse.

2×2: det[abcd]= a×d − b×c Signs for 3×3 row 1: + − +

👆Try [3214] → det = 10 · Try [123456789] in 3×3 — det = 0 (singular!) · Press Show working to see cofactor expansion.

2×2 formula

det[3−416]= 3×6 − (−4)×1 = 18+4 = 22

What det tells you

det ≠ 0 → matrix is invertible

det = 0 → matrix is singular (no inverse)

|det| = area scaling factor of the transformation

💡 Negative det means the transformation flips orientation

🔄

Matrix Inverse

A×A⁻¹ = I. For 2×2: Step 1 find det, Step 2 swap a↔d and negate b,c, Step 3 divide every element by det.

A⁻¹ = (1/det) × [d−b−ca] Only exists when det ≠ 0

MATRIX A — edit values

👆Try [2153] — det=1, clean inverse · Try [4321] — det=−2, fractional inverse · Try singular [1224] — det=0, no inverse exists!

2×2 inverse — 3 steps

Step 1Find det = ad − bc

Step 2Swap a↔d, negate b and c

Step 3Divide every element by det

A=[3−274]: det=26
Swap: [42−73]
A⁻¹=(1/26)[42−73]

3×3 inverse steps

1.Find det(A)
2.Find the 9 minors (2×2 dets)
3.Apply +/− signs → cofactor matrix
4.Transpose → adjugate adj(A)
5.A⁻¹ = adj(A) ÷ det(A)

⚠ det = 0 → NO inverse exists

⚙️

Applications

Where matrices appear in engineering, manufacturing, maintenance and daily life.

⚡Electrical: Circuit analysis — Kirchhoff's laws give a matrix system. Solve [V] = [R][I] to find all currents in a network.
🔧Mechanical: Stiffness matrix — Finite element analysis uses K·u = F where K is the stiffness matrix, u is displacement vector, F is force vector.
🏭Manufacturing: CNC transformation — Rotation and translation of a tool path are applied as matrix multiplications: [x'] = [R][x] + [t].
📡Electrical: Signal processing — The DFT matrix converts a time-domain signal to frequency domain using matrix multiplication.
🚗Mechanical: Robotics — Each joint's rotation and translation is a 4×4 transformation matrix. Multiply them to find end-effector position.
🏗️Structural: Truss analysis — Forces and reactions at nodes form a matrix system. Solve for unknown forces using matrix inversion.
🌡️Maintenance: Data logging — Sensor data stored as matrices. Statistical operations (mean, covariance) computed as matrix operations.
💻Daily life: Computer graphics — Every 3D rotation, scaling or projection you see in games or CAD software is a matrix multiplication.

🎮

Quick Fire Quiz

Test your matrix knowledge — 10 questions.

⚡ Matrix Blitz

Score: 0 Streak: 0 🔥 Q 1/10

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✏

Practice Questions

Show all working. Calculator allowed for checking.

🟢 Tier 1 — Addition & Multiplication

Q1

Given A = [1324] and B = [5102], find: (a) A + B (b) A − B (c) 2A

[6]

(a) [6426] (b) [−4222] (c) [2648]

Q2

Find A × B where A = [3213] and B = [124−6]. Show each cell calculation.

[4]

C₁₁ = 3×1 + 2×4 = 11 C₁₂ = 3×2 + 2×(−6) = −6
C₂₁ = 1×1 + 3×4 = 13 C₂₂ = 1×2 + 3×(−6) = −16
Result = [11−613−16] ✓

Q3

Find A × B where A = [72−9−4] and B = [2369]. Show each cell.

[4]

C₁₁ = 7×2 + 2×6 = 26 C₁₂ = 7×3 + 2×9 = 39
C₂₁ = −9×2 + (−4)×6 = −42 C₂₂ = −9×3 + (−4)×9 = −63
Result = [2639−42−63] ✓

Q4

Multiply A = [103212131] × B = [220132320]. Find C₁₁ and C₂₂.

[4]

C₁₁: Row1·Col1 = 1×2+0×1+3×3 = 11
C₂₂: Row2·Col2 = 2×2+1×3+2×2 = 11

🟣 Tier 2 — Determinants (2×2)

Q5

Find det(A) where A = [5324].

[2]

det(A) = (5×4) − (3×2) = 20 − 6 = 14

Q6

Find det(A) where A = [−3452].

[2]

det(A) = (−3×2) − (4×5) = −6 − 20 = −26

Q7

Find det(A) where A = [6213].

[2]

det(A) = (6×3) − (2×1) = 18 − 2 = 16

🟣 Tier 2 — Determinants (3×3)

Q8

Find det(A) where A = [123014560]. Expand along row 1.

Tip: a row or column with zeros means fewer terms to calculate.

[5]

det(A) = 1(1×0 − 4×6) − 2(0×0 − 4×5) + 3(0×6 − 1×5)
= 1(−24) − 2(−20) + 3(−5) = −24 + 40 − 15 = 1

Q9

Find det(A) where A = [201300511]. Pick the row or column that makes this quickest.

Tip: expand along column 2 — it has two zeros.

[5]

Expanding along column 2 (entries 0, 0, 1) — only the bottom entry contributes:
det(A) = (−1)³ × 1 × det[2130] = −1 × (2×0 − 1×3) = −1 × (−3) = 3

🟠 Tier 3 — Inverses (2×2)

Q10

Find A⁻¹ where A = [3243].

[4]

det(A) = (3×3) − (2×4) = 9 − 8 = 1
A⁻¹ = (1/1) × [3−2−43]
A⁻¹ = [3−2−43]

Q11

Find A⁻¹ where A = [4726].

[4]

det(A) = (4×6) − (7×2) = 24 − 14 = 10
A⁻¹ = (1/10) × [6−7−24]
A⁻¹ = [0.6−0.7−0.20.4]

Q12

Find A⁻¹ where A = [2513]. Verify your answer by checking AA⁻¹ = I.

Tip: multiply your A⁻¹ by A — you should get the identity matrix.

[5]

det(A) = (2×3) − (5×1) = 6 − 5 = 1
A⁻¹ = [3−5−12]
Check: [2513]×[3−5−12] = [1001] ✓

🟠 Tier 3 — Inverses (3×3)

Q13

Find A⁻¹ where A = [211132100]. Use the cofactor / adjugate method.

Steps: (1) det(A), (2) matrix of cofactors, (3) transpose to get adjugate, (4) divide by det(A).

[8]

det(A): expand along row 3 (entries 1, 0, 0) → det(A) = −1
Cofactor matrix C = [02−30−11−1−35]
Adjugate = Cᵀ = [00−12−1−3−315]
A⁻¹ = (1/−1) × adj(A) = [001−2133−1−5]

Q14

Find A⁻¹ where A = [102210121].

det(A) = 7 for this one — your inverse entries will be sevenths.

[8]

det(A) = 7
A⁻¹ = (1/7) × [14−2−2−143−21]
A⁻¹ ≈ [0.1430.571−0.286−0.286−0.1430.5710.429−0.2860.143]