Linear Algebra
Module 2: Matrices Lesson 2 of 3
Previous
Next

Matrix Multiplication

Matrix multiplication is more complex than addition — and more powerful. It represents composing transformations, which is essential for robot kinematics where multiple joints create a chain of transformations.

The Row-Column Rule

To multiply matrices A (size m×n) and B (size n×p), compute each element of the result C (size m×p):

cij=∑k=1naik⋅bkjc_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}

In words: Take row i from A, column j from B, multiply corresponding elements, and sum them.

Step-by-Step Example

[1234][5678]=  ?\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \;?

Element [1,1]: Row 1 of A × Column 1 of B

c11=(1×5)+(2×7)=5+14=19c_{11} = (1 \times 5) + (2 \times 7) = 5 + 14 = 19

Element [1,2]: Row 1 of A × Column 2 of B

c12=(1×6)+(2×8)=6+16=22c_{12} = (1 \times 6) + (2 \times 8) = 6 + 16 = 22

Element [2,1]: Row 2 of A × Column 1 of B

c21=(3×5)+(4×7)=15+28=43c_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43

Element [2,2]: Row 2 of A × Column 2 of B

c22=(3×6)+(4×8)=18+32=50c_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50

Result:

[1234][5678]=[19224350]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Dimension Requirements

For AB to be defined:

  • Number of columns in A must equal number of rows in B
  • If A is m×n and B is n×p, then AB is m×p

Example: (2×3) × (3×4) = (2×4) ✓


Example: (2×3) × (2×4) = undefined ✗

Properties of Matrix Multiplication

Non-Commutative

Generally, AB ≠ BA. Order matters!

[1002][0110]=[0120]\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix} [0110][1002]=[0210]\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix}

Different results!

Associative

(AB)C = A(BC) — you can group multiplications.

Distributive

A(B + C) = AB + AC

Identity Property

AI = IA = A (when dimensions match)

Geometric Interpretation: Transformation Composition

Matrix multiplication represents applying transformations in sequence.

If matrix B represents a transformation (e.g., rotation) and matrix A represents another transformation (e.g., scaling), then AB represents:

  1. First apply B to the input
  2. Then apply A to the result

Reading Order

For AB: Read right to left for transformation order.

  • AB means “do B first, then A”
  • This is the opposite of function notation: f(g(x)) does g first

Robot Kinematic Chain Robotics Application

A 3-joint robot arm has transformations at each joint:

Tend=T1⋅T2⋅T3T_{\text{end}} = T_1 \cdot T_2 \cdot T_3

where:

  • T₁ transforms from base to joint 1
  • T₂ transforms from joint 1 to joint 2
  • T₃ transforms from joint 2 to end-effector

To find where the end-effector is relative to the base, multiply the matrices in order:

pend=T1⋅T2⋅T3⋅plocal\mathbf{p}_{\text{end}} = T_1 \cdot T_2 \cdot T_3 \cdot \mathbf{p}_{\text{local}}

This is forward kinematics — computing end-effector position from joint angles.

Matrix-Vector Multiplication

A special case: multiplying a matrix by a column vector.

[2003][12]=[2×1+0×20×1+3×2]=[26]\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \times 1 + 0 \times 2 \\ 0 \times 1 + 3 \times 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 6 \end{bmatrix}

This transforms the vector (1, 2) into (2, 6).

Velocity Mapping Robotics Application

The Jacobian matrix J maps joint velocities to end-effector velocities:

vend=J⋅q˙\mathbf{v}_{\text{end}} = J \cdot \dot{\mathbf{q}}

where:

  • q˙\dot{\mathbf{q}} is the vector of joint velocities (angular velocities)
  • vend is the Cartesian velocity of the end-effector (linear + angular)

Matrix-vector multiplication transforms joint-space velocities into task-space velocities.

Interactive Visualization

Explore matrix multiplication below. Adjust the matrices and see how transformations compose. The visualization shows how the standard basis vectors (unit square) transform under B, then under A.

Controls

Matrix A

Matrix B

Highlight Element

0.000
01
0.000
01

Matrix Product (AB)

2.000.00
0.002.00
Computing element [1][1]:
(2 × 1 = 2.00)
(0 × 0 = 0.00)
Sum = 2.00

Transformation Composition: Matrix multiplication represents composing transformations. Gray vectors show the original basis. Green shows after applying B, blue shows after applying A to those results. The result AB represents doing B first, then A.

Key Takeaways

  1. Matrix multiplication uses the row-column rule (dot products)
  2. Order matters: AB ≠ BA (non-commutative)
  3. Dimensions must match: (m×n) × (n×p) = (m×p)
  4. Geometric meaning: Composing transformations (do right matrix first)
  5. Robot kinematics: Chain of joint transformations multiplies to give end-effector pose

Practice Problems

Try these mentally or with the interactive tool:

  1. What are the dimensions of (3×5) × (5×2)?
  2. Can you multiply (2×3) × (2×3)? Why or why not?
  3. If A is a 90° rotation and B is a scaling by 2, what does AB do? What about BA?
Answers
  1. (3×2) — 3 rows from first matrix, 2 columns from second
  2. No — 3 columns ≠ 2 rows (inner dimensions don’t match)
  3. AB: scale first, then rotate. BA: rotate first, then scale. Different results! (Scaling changes direction, rotation preserves lengths)

Next Steps

Matrix multiplication is powerful, but not all matrices have inverses. Next, we’ll explore determinants — a number that tells you if a matrix is invertible and reveals geometric properties like area scaling.