Matrices (intro)
The idea
A matrix is a rectangular grid of numbers treated as one object — rows by columns, so a 2×2 matrix holds four entries. Matrices organize linear information: the coefficients of a system of equations, a transformation of the plane, or the connections in a network. Addition and scalar multiplication work entry by entry, just as you would guess; the genuinely new operation is matrix multiplication, whose row-times-column rule is engineered so that multiplying matrices composes the transformations or substitutions they represent.
To multiply, run along a row of the left matrix and down a column of the right, multiplying the pairs and summing: the entry in row i, column j of the product comes from row i of the left and column j of the right. The sizes must cooperate — the left matrix needs as many columns as the right has rows. The misconception that costs the most points is assuming AB = BA; matrix multiplication is not commutative, and swapping the order usually changes every entry. For a 2×2 matrix, the determinant ad − bc is the quick invertibility test: zero means the matrix flattens the plane and cannot be undone.
Worked example
Let A be the 2×2 matrix with rows (2, 1) and (3, 4), and let B be the 2×2 matrix with rows (1, −2) and (0, 5). Compute AB and BA, and find the determinant of A.
- Each entry of AB is a row of A times a column of B. Top row of AB: 2 × 1 + 1 × 0 = 2, then 2 × (−2) + 1 × 5 = −4 + 5 = 1.
- Bottom row of AB: 3 × 1 + 4 × 0 = 3, then 3 × (−2) + 4 × 5 = −6 + 20 = 14. So AB has rows (2, 1) and (3, 14).
- For BA the roles reverse, rows now coming from B: 1 × 2 + (−2) × 3 = −4 and 1 × 1 + (−2) × 4 = −7 give the top row, while 0 × 2 + 5 × 3 = 15 and 0 × 1 + 5 × 4 = 20 give the bottom. BA has rows (−4, −7) and (15, 20).
- Compare: AB and BA disagree in every entry — a vivid reminder that order matters in matrix multiplication.
- Determinant of A: ad − bc = 2 × 4 − 1 × 3 = 8 − 3 = 5. Nonzero, so A is invertible and the system it represents has exactly one solution.
Answer. AB has rows (2, 1) and (3, 14); BA has rows (−4, −7) and (15, 20); det A = 5, so A is invertible.
Check your understanding
- Why is matrix multiplication defined by the row-times-column rule instead of multiplying matching entries?
- What does a determinant of zero mean about the system of equations or the transformation a matrix represents?
- How does writing a linear system as a single matrix equation change the way you think about solving it?
- Why does the order of multiplication matter for matrices when it never matters for ordinary numbers?
Build the foundations first
Matrices (intro) builds on these concepts. If any feel shaky, start there.