The determinant of an identity matrix is always 1, whatever its size. A 2×2, a 5×5 or a 100×100 identity matrix all give exactly 1, because the determinant of any diagonal or triangular matrix is the product of its diagonal entries, and an identity matrix has nothing but 1s on the diagonal. This guide shows the quick reasoning, a few worked examples and the slips that cost marks in exams.
Key takeaways
- det(I) = 1 for every identity matrix I of any order n.
- The reason: the determinant of a triangular matrix is the product of its diagonal entries, and 1 multiplied by itself n times is 1.
- It also follows from the multiplicative rule det(AB) = det(A)det(B).
- It matches the geometric idea that the identity transformation does not change volume or orientation.
- Do not confuse it with the zero matrix, whose determinant is 0.
What is an identity matrix?
An identity matrix, written I or In, is a square matrix with 1s along the main diagonal (top left to bottom right) and 0s everywhere else. Multiplying any compatible matrix by it leaves that matrix unchanged, like multiplying a number by 1. For example, the 3×3 identity matrix is:
[1 0 0; 0 1 0; 0 0 1]
The determinant, in turn, is a single number calculated from a square matrix. It tells you whether the matrix is invertible (non-zero determinant) and how much the associated linear transformation scales area or volume. If you want a general reference on the concept, see Wikipedia’s articles on the identity matrix and the determinant.
Worked examples: 2×2, 3×3 and 4×4
The 2×2 case
For a matrix [a b; c d] the determinant is ad – bc. For the identity, a = 1, b = 0, c = 0 and d = 1, so the determinant is (1)(1) – (0)(0) = 1.
The 3×3 case
Expand along the first row. The determinant is 1 times the determinant of the 2×2 identity formed by deleting row 1 and column 1, minus 0 times something, plus 0 times something else. That gives 1 x 1 = 1.
The 4×4 case
The same pattern repeats. Expanding along the first row, only the top-left entry survives, leaving 1 times the determinant of the 3×3 identity, which is 1. This is a neat example of a recursive argument, and it proves the result for every size by induction.
A short proof that works for any size
- Base case: the 1×1 identity matrix is [1], and its determinant is 1.
- Inductive step: suppose det(In-1) = 1. Expand det(In) along the first row. Every entry in that row is 0 except the first, which is 1.
- The cofactor of that first entry is the determinant of the matrix left after deleting row 1 and column 1, which is In-1.
- So det(In) = 1 x det(In-1) = 1 x 1 = 1.
By induction, det(In) = 1 for all n.
Three other ways to see it
The triangular matrix rule
An identity matrix is both upper and lower triangular. The determinant of a triangular matrix equals the product of its diagonal entries. With n ones, the product is 1.
The multiplicative property
Because I x I = I, we have det(I) = det(I x I) = det(I) x det(I). Let d = det(I). Then d = d squared, so d is 0 or 1. If d were 0, then I would not be invertible, yet I is its own inverse. So d = 1.
The geometric view
In two dimensions, the identity matrix sends the unit square to itself. Its area stays 1 and its orientation is unchanged, so the scaling factor is 1. The same logic applies to volumes in three dimensions and beyond.
Quick reference table
| Matrix | Determinant | Why |
|---|---|---|
| I1 = [1] | 1 | Single entry |
| I2 | 1 | (1)(1) – (0)(0) |
| I3 | 1 | Product of diagonal entries |
| In | 1 | Induction or triangular rule |
| Zero matrix | 0 | All rows are zero |
| 2I3 (twice the identity) | 8 | 2 x 2 x 2 on the diagonal |
The last row shows an easy trap. Scaling an n x n matrix by a number k multiplies its determinant by k to the power n, not by k.
Why it matters
This small fact underpins a lot of linear algebra. It shows that the identity matrix is invertible, since its determinant is not zero. It supports the formula det(A inverse) = 1 / det(A), because A times A inverse equals I and det(I) = 1. It is also the reference point for properties like eigenvalues, where the characteristic polynomial is built from det(A – lambda I). You will meet it in computer graphics, physics, statistics and machine learning whenever matrices describe transformations.
If you enjoy maths puzzles and want another everyday explainer, try our guide on how to find the axis of symmetry of a parabola or the article about why salt is put on icy roads.
Common mistakes in exams
- Writing 0 because most of the entries are 0. The determinant depends on the diagonal product, not on how many zeros there are.
- Forgetting that the matrix must be square before you can take a determinant.
- Mixing up det(kI) with k. For an n x n matrix, det(kI) = k to the power n.
- Applying the 2×2 shortcut (ad – bc) to a 3×3 matrix.
- Dropping the sign pattern when expanding by cofactors.
Frequently asked questions
Is the determinant of an identity matrix always 1?
Yes. For any size n, det(In) = 1.
What is the determinant of the zero matrix?
It is 0, which also means the zero matrix is not invertible.
Does the identity matrix have to be square?
Yes. A determinant is only defined for square matrices, and the identity matrix is square by definition.
What is the determinant of the inverse of the identity?
The identity is its own inverse, so the answer is still 1.
What about a permutation of the identity matrix?
Swapping rows of the identity gives a permutation matrix, whose determinant is +1 or -1 depending on whether the number of swaps is even or odd.
Conclusion
The determinant of an identity matrix is 1 for every size, and you can justify it through induction, the triangular rule, the multiplicative property or simple geometry. Remember the reason rather than just the answer, and the related questions on scaling, inverses and permutations become much easier.
Key takeaways about determinant of an identity matrix
Here is a short summary of determinant of an identity matrix so you can find the main points quickly.
- The determinant of an identity matrix is always 1, whatever its size.
- An identity matrix, written I or In, is a square matrix with 1s along the main diagonal (top left to bottom right) and 0s everywhere else.
- The determinant, in turn, is a single number calculated from a square matrix.
- For a matrix [a b; c d] the determinant is ad – bc.
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For general background, you can look up determinant of an identity matrix on Wikipedia.
