Lesson 13 of 13 · 14 min

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Must-know facts

17 facts

1An m × n matrix has m rows, n columns and mn entries; aᵢⱼ is in row i, column j.
2A matrix with 8 entries can be 1 × 8, 8 × 1, 2 × 4 or 4 × 2.
3There are 2⁹ = 512 matrices of order 3 × 3 with every entry 0 or 1.
4Identity ⊂ scalar ⊂ diagonal ⊂ square: each kind is a special case of the next.
5Equal matrices: same order and equal entries in every position.
6A + B and A − B need the same order; kA scales every entry.
7AB needs (columns of A) = (rows of B); m × n times n × p gives m × p.
8cᵢₖ = row i of A · column k of B.
9AB ≠ BA in general; AB can be defined while BA is not.
10AB = O is possible with A ≠ O and B ≠ O; AB = AC does not give B = C.
11(A + B)² = A² + AB + BA + B²; it equals A² + 2AB + B² only if AB = BA.
12(AB)C = A(BC); A(B + C) = AB + AC; IA = AI = A.
13(A′)′ = A, (kA)′ = kA′, (A + B)′ = A′ + B′, (AB)′ = B′A′.
14Symmetric: A′ = A. Skew symmetric: A′ = −A, diagonal entries all 0.
15A = ½(A + A′) + ½(A − A′): symmetric part plus skew symmetric part.
16A⁻¹ exists only for some square matrices; AA⁻¹ = A⁻¹A = I; it is unique; (AB)⁻¹ = B⁻¹A⁻¹.
17For A = [[3, 1], [−1, 2]], A² − 5A + 7I = O.

Common traps

Where marks are lost

Writing the order of a 2-row, 3-column matrix as 3 × 2.

Rows come first: it is 2 × 3.

Multiplying matrices entry by entry.

Only addition works position by position. A product entry is a row of A run against a column of B, multiplied and summed.

Assuming AB = BA.

Check both products. [[1, 0], [0, −1]] and [[0, 1], [1, 0]] give AB = [[0, 1], [−1, 0]] but BA = [[0, −1], [1, 0]].

Expanding (A + B)² as A² + 2AB + B².

The middle is AB + BA; the shortcut holds only when AB = BA.

Concluding A = O or B = O from AB = O.

[[0, −1], [0, 2]] times [[3, 5], [0, 0]] is O with neither factor zero.

Writing (AB)′ = A′B′ or (AB)⁻¹ = A⁻¹B⁻¹.

Both reverse the order: (AB)′ = B′A′ and (AB)⁻¹ = B⁻¹A⁻¹.

Leaving the constant as a number in a matrix polynomial, as in A² − 5A + 7.

A number cannot be added to a matrix. Write the constant as 7I.

Giving a skew symmetric matrix a non-zero diagonal.

aᵢᵢ = −aᵢᵢ forces every diagonal entry to be 0.

Formulas

10 to know

Order of a product

(m × n)(n × p) = m × p

Inner numbers must match.

Product entry

cᵢₖ = Σⱼ aᵢⱼ bⱼₖ

Row i of A times column k of B, summed.

Sum and scalar multiple

A + B = [aᵢⱼ + bᵢⱼ]; kA = [k aᵢⱼ]

Same order needed for A + B.

Square of a sum

(A + B)² = A² + AB + BA + B²

A² + 2AB + B² only if AB = BA.

Transpose

A = [aᵢⱼ]ₘₓₙ ⇒ A′ = [aⱼᵢ]ₙₓₘ

Rows become columns.

Transpose of a product

(AB)′ = B′A′

Order reverses.

Symmetric–skew split

A = ½(A + A′) + ½(A − A′)

First part symmetric, second skew symmetric.

Inverse

AA⁻¹ = A⁻¹A = I; (AB)⁻¹ = B⁻¹A⁻¹

Square matrices only; unique if it exists.

2 × 2 inverse

[[a, b], [c, d]]⁻¹ = (1/(ad − bc))[[d, −b], [−c, a]]

When ad − bc ≠ 0; proved in the next chapter.

Rotation-type powers

[[cos θ, sin θ], [−sin θ, cos θ]]ⁿ = [[cos nθ, sin nθ], [−sin nθ, cos nθ]]

By induction on n.

Key terms

4 terms

Order: The size m × n of a matrix: m rows by n columns.
Leading diagonal: The entries a₁₁, a₂₂, …, aₙₙ of a square matrix.
Transpose: The matrix obtained by turning the rows of A into columns, written A′ or Aᵀ.
Invertible: A square matrix A with some B of the same order such that AB = BA = I.