Determinants
A single number tells you whether a matrix can be inverted, whether a system has a unique solution and by how much the associated transformation stretches or shrinks areas.
01A number that says whether the system holds up
Every square matrix can be assigned a number, its determinant, which summarizes an essential property: if it is zero, the rows are linearly dependent —some row contributes no new information—, the matrix has no inverse and the associated system either has no solution or has infinitely many. If it is not zero, the opposite holds.
Algebraic: det(A) ≠ 0 is equivalent to A being invertible and to its rank being maximal.
Geometric: its absolute value is the factor by which the associated transformation scales areas
(in the plane) or volumes (in space); the sign tells you whether it preserves or reverses orientation.
Physical: in a circuit, a zero determinant means that the mesh equations are not
independent; it is almost always a sign that the circuit has been set up wrongly.
02How it is computed
Order 2: the difference of the two products
Order 3: Sarrus’ rule
Repeat the first two columns on the right, add the three products of the diagonals that go down to the right and subtract the three products of those that go down to the left.
Any order: cofactor expansion
The minor Mij is the determinant that remains after crossing out row i and column j. The cofactor adds the checkerboard sign:
03Properties, which are shortcuts
| If in the matrix… | the determinant… | What it is good for |
|---|---|---|
| two rows are swapped | changes sign | Reordering the matrix without losing the value, keeping count of the swaps |
| a row is multiplied by k | is multiplied by k | Pulling out common factors. Careful: det(kA) = kn·det(A) for order n |
| a multiple of another row is added to a row | does not change | It is the operation used to triangularize: the key to efficient computation |
| there is a row of zeros | is 0 | Detecting systems without a unique solution without doing the computation |
| two rows are equal or proportional | is 0 | The same |
| the matrix is triangular | is the product of the diagonal | The cheap route: triangularize and multiply |
And two global properties: det(AT) = det(A) —everything said about rows holds for columns— and det(A·B) = det(A)·det(B), from which it follows that det(A−1) = 1 / det(A).
det(A + B) ≠ det(A) + det(B), except by coincidence. It behaves well with the product, not with the sum.
04Seeing it work
The matrix below can be edited and operated on. Each row operation shows what happened to the determinant, and the computation appears by Sarrus’ rule and by cofactors at the same time.
By Sarrus’ rule
By cofactors of the first row
05Adjugate matrix, inverse and Cramer’s rule
The adjugate matrix adj(A) is built by replacing each element with its cofactor and transposing. With it, the inverse has a closed formula:
Cramer’s rule solves a system of n equations in n unknowns with a nonzero determinant: each unknown is a quotient of determinants, where the numerator is obtained by replacing the column of that unknown with the column of constant terms.
A two-mesh circuit leads to 6·I₁ − 2·I₂ = 10 and −2·I₁ + 8·I₂ = 0 (in volts and ohms).
det(A) = 6·8 − (−2)(−2) = 44. Replacing the first column: det(A₁) = 10·8 − (−2)·0 = 80, and the second: det(A₂) = 6·0 − 10·(−2) = 20.
I₁ = 80/44 = 1.82 A and I₂ = 20/44 = 0.45 A. Check in the second equation: −2(1.818) + 8(0.4545) = −3.64 + 3.64 = 0. ✔
For n unknowns you have to compute n + 1 determinants. With n = 3 it is convenient and can be done by hand; with n = 20 it is impractical, and that is where Gaussian elimination is used, which costs on the order of n³ operations. Circuit simulation programs never use Cramer.
06What it means geometrically
The columns of a 2 × 2 matrix are two vectors in the plane. The determinant is, in absolute value, the area of the parallelogram they form; for 3 × 3, the volume of the parallelepiped of the three columns (the scalar triple product).
That is why det = 0 reads as “the transformation flattens space”: it goes from two dimensions to a line, or from three to a plane. And since flattening loses information, there is no way back: there is no inverse.
07Determinants in electronics
- Solving meshes and nodes. With two or three unknowns, Cramer gives the result without solving for anything step by step, and shows where each term comes from.
- The characteristic determinant. When analyzing a circuit with inductors and capacitors in the complex frequency domain, the roots of the system determinant are the poles of the response: they tell you whether the circuit oscillates, whether it is stable and at what natural frequency. It is the gateway to eigenvalues and eigenvectors.
- Two-port networks. The determinant of the ABCD parameter matrix is 1 for a passive and reciprocal network: it is an immediate check that the parameters have been computed correctly.
- Stability. The Routh-Hurwitz criterion, which in control systems decides whether a feedback loop is stable, is written as a sequence of determinants that must all be positive.
08In the lab
For a 3 × 3 matrix with at least one zero, compute the determinant by Sarrus’ rule, by cofactors along the row with the zero and by triangularizing. All three have to agree; write down how many multiplications each one took.
Build a three-mesh circuit with your own values, set up the system, solve it by Cramer’s rule and check each current with Kirchhoff’s voltage law. Then compare with a simulation.
In a 3 × 3 matrix where one element is the parameter k, set det = 0 and solve for k. Interpret it: for that value the system stops having a unique solution. It is the same setup used later to compute eigenvalues.
09Common mistakes
- Using Sarrus’ rule on 4 × 4. It does not exist; you have to expand by cofactors or triangularize.
- Forgetting the cofactor sign (−1)i+j, especially at position (1,2).
- Believing that det(kA) = k·det(A): for order n it is kn·det(A), because all n rows are multiplied.
- Computing determinants of non-square matrices. They are not defined.
- Applying Cramer with a zero determinant: there the method says nothing and you have to analyze the system using the rank.
- Losing the sign when swapping rows while triangularizing.
10Self-assessment
Compute the determinant of [[2, 3], [4, 5]].
2·5 − 3·4 = 10 − 12 = −2.
If det(A) = 5 with A of order 3, what is det(2A)?
2³ · 5 = 40: each of the three rows was multiplied by 2.
A 4 × 4 upper triangular matrix has 2, −1, 3 and 5 on the diagonal. What is its determinant?
The product of the diagonal: 2·(−1)·3·5 = −30.
What happens to the determinant if twice row 1 is added to row 3?
Nothing: it does not change. That is why this is the operation used to triangularize.
If det(A) = 0, what can be said about the system A·x = b?
That it has no unique solution: either it has none, or it has infinitely many. Which of the two it is is decided by comparing the rank of A with that of the augmented matrix.
det(A) = 3 and det(B) = −2, with A and B of order 3. What are det(A·B) and det(A−1)?
det(A·B) = 3·(−2) = −6 and det(A−1) = 1/3.
11Further reading
- Stanley I. Grossman. Álgebra lineal. 6th ed., McGraw-Hill, 2008 (in Spanish). The chapter on determinants develops the properties with accessible proofs and many exercises.
- Howard Anton. Elementary Linear Algebra. 5th ed. (Spanish edition, “Introducción al álgebra lineal”), Limusa Wiley, 2011. It explains cofactor expansion and Cramer’s rule with care, with applications to circuits.
- Gilbert Strang. Linear Algebra and Its Applications. 4th ed. (Spanish edition, “Álgebra lineal y sus aplicaciones”), Cengage Learning, 2007. Its chapter 4 is the one that best connects the determinant with volume and with the cost of computing it.