Matrices, Determinants & Inverses — Maths for Quantum Chemistry

Matrices transform space

The bookkeeping of symmetry, linear systems, and spectroscopy

A matrix is a rectangular grid of numbers that acts on vectors — rotating, reflecting, stretching or shearing them. In chemistry, every symmetry operation (rotation, reflection, inversion) is exactly such a matrix, and the machinery of group theory manipulates them to predict which vibrational modes are IR-active and which spectral transitions are allowed.

The single most important number attached to a square matrix is its determinant. For a 2×2 matrix the formula is:

det (a c b d) = a d - b c

The determinant measures the area-scaling factor of the transformation. When it is zero the matrix squashes space onto a line and has no inverse; otherwise the inverse exists and is given by:

(a c b d)^{- 1} = \frac{1}{a d - b c} (d - c - b a)

Matrix multiplication combines transformations: applying $B$ then $A$ is written $A B$ (note the order matters — multiplication is generally not commutative). Two special matrices recur constantly: the identity $I$ (diagonal ones, off-diagonal zeros) leaves every vector unchanged, and the transpose $A^{T}$ swaps rows and columns.

Key formulas at a glance

Determinant (2×2)

det (A) = a d - b c

Inverse (2×2)

A^{- 1} = \frac{1}{d e t A} (d - c - b a)

Identity matrix

I = (1001), A I = I A = A

Transpose

(A^{T})_{ij} = A_{ji}; (A B)^{T} = B^{T} A^{T}

Proper rotation (C₂)

(- 1 0 0 - 1), det = + 1

Reflection (σ_xz)

(10 0 - 1), det = - 1

Non-commutativity is often the first surprise. Because matrix multiplication performs one transformation then another, the order matters: in general $A B \neq = B A$ . In group theory this is the distinction between Abelian and non-Abelian groups — e.g., $C_{3 v}$ (ammonia) is non-Abelian because $C_{3} σ_{v} \neq = σ_{v} C_{3}$ .

The trace of a matrix (sum of its diagonal entries) is a similarity invariant: applying a change-of-basis transformation $P$ gives $tr (P A P^{- 1}) = tr (A)$ . In group theory the trace of a representation matrix is its character, $χ$ . Because the character does not change when we switch from one set of basis functions to another, it is the quantity tabulated in character tables and used to work out reducible vs irreducible representations, selection rules, and molecular orbital symmetries.

The C₂v symmetry operations — a 3×3 representation

E (identity)

100010001, χ = 3

C₂ (about z)

- 1 00 0 - 1 0 001, χ = - 1

σᵥ (xz plane)

100 0 - 1 0 001, χ = 1

σᵥ′ (yz plane)

- 1 00 010001, χ = 1

Common pitfalls

Element-wise multiplication instead of row × column. The entry $(A B)_{ij} = \sum_{k} A_{ik} B_{kj}$ — you must take the dot product of row $i$ of $A$ with column $j$ of $B$ , not just multiply matching entries.
Ignoring order in matrix multiplication. In C₂v, $C_{2} σ_{x z} = σ_{yz}$ but $σ_{x z} C_{2} = σ_{yz}$ — for this particular pair the result happens to agree; try $C_{3} σ_{v}$ in ammonia to see a genuine non-commutative case.
Trying to invert a singular matrix. If $det A = 0$ there is no inverse; the system of equations has either no solution or infinitely many. Always compute the determinant before attempting to invert.
Sign errors in the 3×3 cofactor expansion. The cofactor signs follow the checkerboard pattern $+ - + - + - + - +$ , so the second cofactor picks up a minus sign.

Symmetry & beyond

Every symmetry operation of a molecule (rotation, reflection, inversion) is a matrix, and group theory manipulates them to predict spectroscopy and bonding. Solving linear systems — such as simultaneous Beer–Lambert equations for a two-component mixture — is also a matrix problem. Open the transformation lab to see the determinant as a living area.

Going deeper

The group multiplication table of C₂v (E, C₂, σᵥ, σᵥ′) encodes the closure axiom: every product of two operations is again an operation in the group. Check for instance that $C_{2} \cdot C_{2} = E$ (the 3×3 matrices square to the identity) and that $σ_{v} \cdot σ_{v}^{'} = C_{2}$ — as you can verify by direct matrix multiplication. This algebraic structure is what allows chemists to classify molecular orbitals and predict which IR and Raman bands are symmetry-allowed without ever solving the full Schrödinger equation.

Transformation lab

Drag the four matrix entries and watch the unit square deform. The shaded area is |det| — the orientation flips when det goes negative.

The matrix transforms the unit square — its determinant is the area

det (10 0.5 1) = 1.00

b0.5

determinant1.00

area scaling×1.00

orientationpreserved

The red and green arrows show where the basis vectors land. The shaded area is |det|; when det = 0 the square collapses to a line (no inverse). Reflections give det = −1.

Worked example 1C₂ rotation matrix acting on a water-molecule coordinate

A C₂ rotation about the $z$ -axis maps $(x, y) \to (- x, - y)$ . Its matrix is $C_{2} = (- 1 0 0 - 1)$ .

Apply it to the oxygen lone-pair position vector $r = (1, 0.5)$ and verify that a second C₂ returns the vector to its starting point.

Worked example 22×2 matrix multiplication: combining symmetry operations

The reflection $σ_{x z}$ maps $y \to - y$ and the inversion $i$ maps $(x, y) \to (- x, - y)$ (in 2-D). Write each as a 2×2 matrix and compute $i σ_{x z}$ . What operation does the product represent?

Worked examples

Determinants, inverses, and a two-component Beer–Lambert system.

Worked example 3Determinant and inverse of a 2×2 matrix

Find the determinant and the inverse of $A = (3214)$ .

Worked example 4Solving a two-component Beer–Lambert mixture

A mixture of compounds X and Y is measured at two wavelengths. At 400 nm: $ε_{X} = 1200, ε_{Y} = 300$ (L mol⁻¹ cm⁻¹). At 550 nm: $ε_{X} = 200, ε_{Y} = 800$ . The absorbances in a 1.00 cm cell are $A_{400} = 0.540$ and $A_{550} = 0.320$ .

Find the concentrations $c_{X}$ and $c_{Y}$ .

Worked example 53×3 determinant: evaluating a symmetry matrix

The 3-D inversion matrix is $i = - 1 00 0 - 1 0 00 - 1$ . Evaluate its determinant by cofactor expansion along the first row.

Worked example 6Solving a 3-unknown mass-balance by Cramer's rule

A Hess's-law cycle for the formation of SO₃ involves three intermediate reactions with unknown enthalpy contributions $x_{1}, x_{2}, x_{3}$ (kJ mol⁻¹) satisfying:

101110011 x_{1} x_{2} x_{3} = - 297 - 99 - 495

Find $x_{1}, x_{2}, x_{3}$ by evaluating the 3×3 determinant and applying Cramer's rule.

Worked example 7Non-commutativity: AB ≠ BA for rotation and reflection

Let $R = (01 - 1 0)$ (90° rotation) and $S = (10 0 - 1)$ (reflection in the $x$ -axis). Compute both $RS$ and $SR$ and confirm they are different.

ChallengeC₂v representation matrices — closure and the group table

The C₂v point group has four symmetry operations: $E, C_{2}, σ_{v} (x z), σ_{v}^{'} (yz)$ . Their 3×3 representation matrices (acting on the Cartesian coordinates $(x, y, z)$ ) are:

E = 100010001, C_{2} = - 1 00 0 - 1 0 001

σ_{v} = 100 0 - 1 0 001, σ_{v}^{'} = - 1 00 010001

(a) Compute the trace (character) of each matrix. (b) Show that $σ_{v} \cdot σ_{v}^{'} = C_{2}$ by direct matrix multiplication, verifying closure. (c) Confirm that the group is closed by listing all products from the group table.

Check yourself

Four questions on determinants, inverses and symmetry operations.

Question 1 of 4 · Score 0

The determinant of the 2×2 matrix [[a, b], [c, d]] is:

det = a d - b c

Choose an answer.