Source: Friedberg, Insel, Spence – Linear Algebra, 4th Ed.
Tags: determinant, 2x2 matrix, det(A), ad minus bc, invertibility, parallelogram area, orientation, right-handed coordinate system, cofactor, linear algebra
Difficulty: Introductory–Intermediate | Prerequisites: Matrix operations, invertibility, fields (Chapter 3 basics).
Determinants are scalar-valued functions defined on square matrices. They crop up everywhere in linear algebra, from testing whether a matrix is invertible to computing eigenvalues (Chapter 5) and measuring geometric quantities like area and volume. This section starts small, with 2×2 matrices, and builds the intuition you will need before the general n×n case in Section 4.2. If you are comfortable with basic matrix arithmetic from Chapter 3, you are ready.
The determinant of a 2×2 matrix [[a, b], [c, d]] is the scalar ad − bc. It tells you whether the matrix is invertible (nonzero determinant means yes) and, geometrically, its absolute value equals the area of the parallelogram spanned by the matrix's row vectors.
Determinant of a 2×2 matrix
For A = [[a, b], [c, d]] with entries from a field F, det(A) = ad − bc. Also written |A|. In simple terms, multiply the diagonals and subtract: top-left × bottom-right minus top-right × bottom-left.
n-linearity (2-linearity in this section)
The determinant is a linear function of each row when the other row is held fixed. In simple terms, if you scale one row or add two rows together (keeping the other row untouched), the determinant responds linearly.
Orientation of an ordered basis
For an ordered basis {u, v} of R², the orientation O(u, v) = det([u; v]) / |det([u; v])|. It equals +1 or −1. Think of it as: does your basis spin counterclockwise (right-handed, +1) or clockwise (left-handed, −1)?
Right-handed coordinate system
An ordered basis {u, v} where u can be rotated counterclockwise through an angle θ (0 < θ < π) to align with v. The standard basis {e₁, e₂} is right-handed. Orientation = +1.
Left-handed coordinate system
The opposite case: u must rotate clockwise to align with v. Orientation = −1.
Parallelogram determined by u and v
The parallelogram with adjacent sides u and v, both emanating from the origin. If u and v are parallel (linearly dependent), the "parallelogram" collapses to a line segment with area zero.
For A = [[a, b], [c, d]], det(A) = ad − bc.
Example: A = [[1, 2], [3, 4]] gives det(A) = (1)(4) − (2)(3) = −2.
Example: B = [[3, 2], [6, 4]] gives det(B) = (3)(4) − (2)(6) = 0.
det(A + B) ≠ det(A) + det(B) in general.
From the examples above, A + B = [[4, 4], [9, 8]], so det(A + B) = 32 − 36 = −4, while det(A) + det(B) = −2 + 0 = −2.
This is a common source of confusion. The determinant is linear in each row separately, not in the matrix as a whole.
The determinant is a linear function of each row when the other row is held fixed.
Concretely: det([u + kv; w]) = det([u; w]) + k · det([v; w]).
The same holds for the second row: det([w; u + kv]) = det([w; u]) + k · det([w; v]).
This property is sometimes called "multilinearity" (here, 2-linearity).
det(A) ≠ 0 if and only if A is invertible.
When A is invertible, the inverse has a clean formula:
A⁻¹ = (1 / det(A)) · [[A₂₂, −A₁₂], [−A₂₁, A₁₁]].
If det(A) = 0, the matrix is singular (not invertible), its rows are linearly dependent, and its rank is less than 2.
The area of the parallelogram determined by vectors u and v equals |det([u; v])|.
More precisely: Area = O(u, v) · det([u; v]), and taking absolute values gives the area.
Example: u = (−1, 5), v = (4, −2). Area = |det([[−1, 5], [4, −2]])| = |(−1)(−2) − (5)(4)| = |2 − 20| = 18.
O(u, v) = +1 when {u, v} is right-handed (counterclockwise rotation from u to v).
O(u, v) = −1 when {u, v} is left-handed (clockwise rotation).
By convention, O(u, v) = +1 when {u, v} is linearly dependent (the degenerate case).
2×2 Determinant:
det([[a, b], [c, d]]) = ad − bc
2×2 Inverse (when det ≠ 0):
A⁻¹ = (1 / det(A)) · [[d, −b], [−c, a]]
Parallelogram area:
Area = |det([u; v])| where u, v are row vectors forming the matrix.
The 2×2 determinant shows up whenever you need to check whether two vectors in the plane are linearly independent, for instance when verifying that a coordinate transformation is valid. In computer graphics, determinants test whether transformations preserve or reverse orientation (mirroring), and the absolute value of the determinant gives the scaling factor applied to areas.
Students often assume det(A + B) = det(A) + det(B). It does not. The determinant is multilinear in the rows, not additive in the matrices.
The area of a parallelogram is |det|, not det itself. The determinant can be negative; the area cannot.
A zero determinant does not mean "the matrix is zero." It means the rows are linearly dependent (the matrix is singular).
The inverse formula A⁻¹ = (1/det(A)) · [[d, −b], [−c, a]] only works for 2×2 matrices. Do not try to generalise this layout to larger matrices directly.
⚠️ The formula det(A) = ad − bc is tested constantly, both on its own and as a sub-step inside larger determinant computations (cofactor expansion reduces everything to 2×2 determinants eventually).
⚠️ Theorem 4.2 (det ≠ 0 ⟺ invertible) is one of the most important equivalences in the course. Expect true/false questions and proof-based questions on this.
⚠️ The geometric interpretation (area, orientation) appears in applied problems and occasionally in proofs. Know that |det| = area and that the sign encodes orientation.
⚠️ Row linearity (Theorem 4.1) is the conceptual foundation for everything in Sections 4.2–4.5. Understand the statement, even if you are not asked to reproduce the proof.
True or false: det: M₂ₓ₂(R) → R is a linear transformation.
Fill in the blank: det([[5, 3], [2, 1]]) = ___.
True or false: If det(A) = 0 for a 2×2 matrix A, then A is invertible.
The area of the parallelogram determined by u = (1, 3) and v = (−3, 1) is ___.
True or false: Swapping the two rows of a 2×2 matrix changes the sign of its determinant.
(Answers: 1. False. 2. 5·1 − 3·2 = −1. 3. False, det = 0 means not invertible. 4. |1·1 − 3·(−3)| = |1 + 9| = 10. 5. True.)
Q: Compute det(A) for A = [[6, −3], [2, 4]].
A: det(A) = (6)(4) − (−3)(2) = 24 + 6 = 30.
Q: Let A = [[a, b], [c, d]] with det(A) ≠ 0. Write the formula for A⁻¹.
A: A⁻¹ = (1/(ad − bc)) · [[d, −b], [−c, a]].
Q: If B is obtained by interchanging the rows of a 2×2 matrix A, what is det(B) in terms of det(A)?
A: det(B) = −det(A).
Q: Prove that if the two columns of A ∈ M₂ₓ₂(F) are identical, then det(A) = 0.
A: If both columns are the same, say A = [[a, a], [c, c]], then det(A) = ac − ac = 0.
Q: True or false – the area of the parallelogram spanned by u and v equals det([u; v]).
A: False. The area equals |det([u; v])|. The determinant itself can be negative.
This section sets up the general n×n determinant in Section 4.2 by establishing the key properties (row linearity, the invertibility criterion, and the geometric interpretation) in the simplest possible case. The invertibility criterion (det ≠ 0 ⟺ invertible) extends to n×n matrices and connects directly to the rank theorems in Chapter 3. The determinant's role in computing eigenvalues begins in Chapter 5, where det(A − λI) = 0 defines the characteristic polynomial.
determinant, 2x2 determinant, ad minus bc, matrix inverse formula, invertible matrix test, parallelogram area, orientation of basis, right-handed system, left-handed system, multilinearity, 2-linearity, scalar-valued function on matrices, cofactor, det(A), linear algebra chapter 4, Friedberg Insel Spence