Sum of Determinants Calculator
Enter the entries of two square matrices (2 × 2 or 3 × 3) and this calculator finds each determinant and their sum |A| + |B| instantly, with step-by-step formulas explained.
Determinant Sum Solver
Sum of Determinants – Calculation Example
Sum of Determinants Formula
Where:
- \(A\) and \(B\) — square matrices of the same size (2 × 2 or 3 × 3)
- \(\det(A)\), written \(|A|\) — the determinant of matrix \(A\)
- \(\det(B)\), written \(|B|\) — the determinant of matrix \(B\)
The first line states the operation itself: each determinant is computed independently, and the two scalar results are then added together. The second and third lines show how a single determinant is evaluated for 2 × 2 and 3 × 3 matrices (cofactor expansion along the first row).
Sum of Determinants – Calculation Example
Example: two 3 × 3 matrices with the calculator's default values:
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
\[ \det(A) = 1(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0 \]
\[ \det(B) = 1(1 \cdot 1 - 0 \cdot 0) - 0(0 \cdot 1 - 0 \cdot 0) + 0(0 \cdot 0 - 1 \cdot 0) = 1 \]
\[ |A| + |B| = 0 + 1 = 1 \]
Result: \(\det(A) = 0\), \(\det(B) = 1\), and the sum of the determinants is \(1\). Note that matrix \(A\) is singular (its rows are linearly dependent), which is why its determinant is zero.
How to Use the Calculator
- Select the Matrix Size — either 2 × 2 or 3 × 3 — from the dropdown at the top. The 3 × 3 fields appear or hide automatically.
- Enter the entries of Matrix A in the fields labeled a₁₁ through a₃₃ (row by row).
- Enter the entries of Matrix B in the fields labeled b₁₁ through b₃₃.
- The results update instantly: |A|, |B|, and the sum |A| + |B|.
What Is the Sum of Determinants?
A determinant is a single number computed from a square matrix. It encodes key properties of the linear transformation the matrix represents: whether the transformation is invertible, how it scales areas or volumes, and how it orients space. A determinant of zero means the matrix is singular — it collapses space and has no inverse.
The sum of determinants is simply the scalar addition \(\det(A) + \det(B)\). Each matrix is evaluated on its own, and only the two resulting numbers are combined. This is a common exercise in linear algebra courses and appears in proofs and formulas where determinants of several matrices must be pooled — for example, when checking linear dependence, evaluating block-matrix expressions, or combining characteristic polynomials.
A crucial point: the sum of determinants is generally not equal to the determinant of the sum. In symbols, \(\det(A + B) \neq \det(A) + \det(B)\) in almost all cases. The determinant is a nonlinear, multiplicative function — it satisfies \(\det(AB) = \det(A)\,\det(B)\), which you can verify with our Determinant Multiplication calculator — but it does not distribute over matrix addition. If you need the determinant of \(A + B\), first add the matrices element by element with the Matrix Addition calculator, then compute the determinant of the result.
Interpreting the output is straightforward: |A| and |B| are the two individual determinants, and |A| + |B| is their ordinary arithmetic sum, which can be positive, negative, or zero. A zero sum does not imply either matrix is singular — one determinant can simply cancel the other.
Assumptions, Limitations and Tips
- Same size, square matrices only. Determinants are defined only for square matrices, and this tool accepts 2 × 2 and 3 × 3 inputs. The two matrices do not need to be related in any other way.
- Signs matter. The determinant can be negative. Swapping two rows or two columns flips the sign, so double-check the order in which you enter values (row first, then column: a₂₁ is row 2, column 1).
- Round-off. The calculator rounds the displayed results, so very large entries or numbers with many decimals may lose a little precision in the shown output.
- Singular matrices. If a determinant comes out as 0, the matrix has no inverse and its rows (and columns) are linearly dependent. To explore such a matrix further, try the Rank of 3×3 Matrix calculator or the Inverse Matrix 3×3 calculator.
- Need subtraction instead? For the difference \(\det(A) - \det(B)\), you can enter \(-B\) as the second matrix, since \(\det(-B) = -\det(B)\) for 3 × 3 (and, more generally, \(\det(-B) = (-1)^n\det(B)\) for an \(n \times n\) matrix). Element-by-element matrix subtraction is covered by our Subtraction of Two Matrices calculator.
Frequently Asked Questions
Is the sum of determinants the same as the determinant of the sum?
No, in almost all cases \(\det(A+B) \neq \det(A) + \det(B)\). For example, with \(A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\), you get \(\det(A) + \det(B) = 1 - 1 = 0\), but \(\det(A+B) = \det\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} = 0\) is coincidental here — change one entry and the two values diverge. To get \(\det(A+B)\), add the matrices first, then take one determinant.
What is the fastest way to compute a 3 × 3 determinant by hand?
Many people use the Sarrus rule: copy the first two columns to the right of the matrix, then add the three down-right diagonal products and subtract the three up-right diagonal products. For \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\) this gives \(aei + bfg + cdh - ceg - bdi - afh\). It matches cofactor expansion along the first row and works only for 3 × 3 matrices, not larger ones.
Can the sum be zero even if both matrices are invertible?
Yes. The sum \(\det(A) + \det(B) = 0\) only tells you the two determinants are opposites, for example \(5\) and \(-5\). Both matrices can be perfectly invertible. Only an individual determinant of zero signals a singular (non-invertible) matrix — the sum alone says nothing about invertibility of either matrix.
What does a negative determinant mean?
The sign of a determinant encodes orientation. Geometrically, \(|\det(A)|\) is the factor by which the matrix scales area (2 × 2) or volume (3 × 3), while a negative value means the transformation flips or mirrors space — like reflecting in a mirror. A negative determinant is perfectly normal and does not indicate an error in your calculation.
Do the two matrices have to be the same size?
Mathematically, no — you can add the determinants of a 2 × 2 and a 3 × 3 matrix, since they are just two numbers. However, this calculator requires both matrices to share the selected size (2 × 2 or 3 × 3) for a consistent display. If you need mixed sizes, compute each determinant separately and add the results by hand.
Can I enter decimals or negative numbers?
Yes, both are fully supported. Determinants of matrices with decimal or negative entries are computed the same way, and the final sum may be positive, negative, or zero. Just note that the displayed results are rounded, so entries with many decimal places or very large magnitudes may show a slightly rounded value in the output.
Why did my answer come out wrong when I calculated by hand?
The most common mistakes are transposing entries (entering \(a_{12}\) where \(a_{21}\) belongs — remember the first index is the row) and sign slips in cofactor expansion, where the middle term of the first row is subtracted. Also double-check that you expanded along the correct row and kept the alternating \(+,-,+\) pattern. If the matrix has a row that is a multiple of another row, a determinant of 0 is expected, not an error.