Math 115. Basic Information on Matrices

This handout gives some basic information on matrices.

If you’ve already taken Math 54 (which is not a prerequisite for Math 115), then you probably don’t need to read this.

2 × 2 Matrices First, we note that a 2 × 2 matrix is an object

[ \begin{matrix}a & b \\ c & d \end{matrix} ]  (1)

with a, b, c, d ∈ ℝ. The entries of the matrix are the numbers a, b, c, d. We define matrix multiplication by the formula

[ \begin{matrix}a & b \\ c & d \end{matrix} ] [ \begin{matrix}e & f \\ g & h \end{matrix} ] = [ \begin{matrix}ae + bg & af + bh \\ ce + dg & cf + dh \end{matrix} ] .  (2)

Lemma. Multiplication of 2 × 2 matrices, as in (2), is associative.

Proof. To each 2 × 2 matrix as in (1), we associate a function ϕ: ℝ^{2} → ℝ^{2} given by ϕ(x, y) = (ax + by, cx + dy) . (Not all functions ℝ^{2} → ℝ^{2} come from matrices in this way, but all 2 × 2 matrices determine functions ℝ^{2} → ℝ^{2} , so this gives an injection from the set of 2 × 2 matrices to the set of functions ℝ^{2} → ℝ^{2} .)

One can then check by direct computation that if ψ: ℝ^{2} → ℝ^{2} is similarly asso-

ciated to the matrix [ \begin{matrix}e & f \\ g & h \end{matrix} ], then the matrix product as in (2) is associated to the

composition ϕ ∘ ψ. Since function composition is associative, it follows that matrix multiplication is also associative. □

One also has the 2 × 2 identity matrix I_{2} = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ], and it is easy to check by

computation that I_{2}A = AI_{2} = A for all 2 × 2 matrices A. (Also, I_{2} is associated to the identity function ℝ^{2} → ℝ^{2} .)

A 2 × 2 matrix A has a determinant, denoted det A or |A|, given by

det [ \begin{matrix}a & b \\ c & d \end{matrix} ] = | \begin{matrix}a & b \\ c & d \end{matrix} | = ad − bc .

(One should be careful not to confuse the notation for determinant with the notation for absolute value.)

We have det(AB) = (det A)(det B) for all 2 × 2 matrices. (This can be checked by direct computation.)

A 2 × 2 matrix A is said to be nonsingular if det A ≠ 0 .

If A is nonsingular, then it has an inverse A^{−1} , given by the formula

[ \begin{matrix}a & b \\ c & d \end{matrix} ]^{−1} = [ \begin{matrix}\frac{d}{det A} & −\frac{b}{det A} \\ −\frac{c}{det A} & \frac{a}{det A} \end{matrix} ] .

One can check (again by computation) that AA^{−1} = A^{−1}A = I_{2} if A is nonsingular.

Matrices of Some Other Sizes

Similarly, we can define 2 × 1 matrices, 1 × 2 matrices, and 1 × 1 matrices to be matrices of the form

[ \begin{matrix}a \\ b \end{matrix} ] ,  [ a b] ,  and [ a] ,

respectively, and one can define multiplication of such matrices in certain cases:

[ \begin{matrix}a & b \\ c & d \end{matrix} ] [ \begin{matrix}e \\ f \end{matrix} ] = [ \begin{matrix}ae + bf \\ ce + df \end{matrix} ] ,  [ a b] [ \begin{matrix}c & d \\ e & f \end{matrix} ] = [ ac + be ad + bf ] ,  and

[ a b] [ \begin{matrix}c \\ d \end{matrix} ] = [ ac + bd] .

As in the earlier lemma, these multiplication operations are associative (when defined). Note that a product of an m × n matrix with an m′ × n′ matrix is defined if and only if n = m′ (and, if it is defined, then the product is an m × n′ matrix). This corresponds to the fact that an m × n matrix can be associated to a certain type of function ℝ^{n} → ℝ^{m}, and that a composition ϕ ∘ ψ is defined if and only if the domain of ϕ equals the codomain of ψ.

Tranposes of Matrices

The transpose A^{t} of a matrix A is defined to be the matrix obtained by interchanging the rows and columns of the matrix:

[ \begin{matrix}a & b \\ c & d \end{matrix} ]^{t} = [ \begin{matrix}a & c \\ b & d \end{matrix} ] ,  [ \begin{matrix}a \\ b \end{matrix} ]^{t} = [ a b] ,  and [ a b]^{t} = [ \begin{matrix}a \\ b \end{matrix} ] .

Note that (A^{t})^{t} = A for all A, and that A^{t}B^{t} = (BA)^{t} (whenever the product AB is defined).

Also, if A is a 2 × 2 matrix, then det(A^{t}) = det A and (A^{t})^{−1} = (A^{−1})^{t}.