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Matrix Multiplication · A Tutorial
Tutorial

Matrix
multiplication.

Four rounds, three quick lessons. You'll learn to read matrices, pair, multiply, and compute — then open the Matrix Workshop for determinants, inverses, simultaneous equations and transformations.

01 Pair and multiply 1×2 · 2×1
02 Add the products 2×2 · 2×1
03 Full multiplication 2×2 · 2×2
04 Mix and finish two problems · type the result
Matrix Workshop — CSEC Classes 17–19

Already know how to multiply? Jump into the exam work: determinants and inverses, solving simultaneous equations with matrices, and transformations on the grid — every step written out.

Matrix
sizes.

A matrix is named by its dimensions — rows × columns. Some sizes have special names. Scroll through a few.

Square matrix n × n Same number of rows and columns. 2×2 and 3×3 are square.
Row matrix 1 × n A single row. Like 1×2 or 1×4.
Column matrix n × 1 A single column. Like 2×1 or 3×1.
Quick check — what size is this matrix?
→
×
01 / 04 A 1×2 · B 2×1

Can these
multiply?

Not every pair of matrices can be multiplied. Use this shortcut to know if you can.

(m × n) (p × q)
The inner dimensions must match.
When they do, the result is the outer pair: m × q.
Try a few:

What size
will the answer be?

If A and B can multiply, the answer takes the outer dimensions of the pair.

(m × n) (n × q) → m × q
Take the outer dimensions and you have the size of A · B.
Try predicting the result size for each pair of matrices.
Tutorial complete

Well
done.

You can read sizes, check compatibility, predict the result, and compute it.
That's matrix multiplication.

Matrix Workshop · Class 17

Determinant
& inverse.

Type any 2×2 matrix. The determinant, adjoint and inverse are worked line by line — fractions stay as fractions.

Matrix A
A =

For A = abcd: det A = ad − bc, adj A swaps a and d and changes the signs of b and c, and A⁻¹ = (1/det A) × adj A.

Matrix Workshop · Class 18

Solve with
matrices.

Write the pair as AX = B, then X = A⁻¹B. Keep 1/det outside until the very end — that is the shortcut.

The equations
x + y =
x + y =

Type a negative coefficient for a minus sign: 2x + (−3)y means 2x − 3y.

Try a past-paper pair
Matrix Workshop · Class 19

Transformation
matrices.

A 2×2 matrix moves every point of the object. Multiply each vertex, plot the image, then undo it with the inverse.

Matrix T
T =
Object triangle ABC
A(,)
B(,)
C(,)
Given an image point, find where it came from.
P′(,)
object ABCimage A′B′C′