Determinant
-2.000000
Size
2x2
Operation
Determinant
| Determinant | -2.000000 |
| Invertible | Yes |
| Size | 2x2 |
det(A) = ad − bcCalculates the determinant of a 2×2 matrix [[a,b],[c,d]] by multiplying the main diagonal and subtracting the anti-diagonal product.
Where:
a, d= Main diagonal elements (top-left, bottom-right)b, c= Anti-diagonal elements (top-right, bottom-left)det = a(ei−fh) − b(di−fg) + c(dh−eg)Expands along the first row using cofactors for a 3×3 matrix [[a,b,c],[d,e,f],[g,h,i]].
Where:
a,b,c= First row elementsd,e,f= Second row elementsg,h,i= Third row elementsC[i][j] = ∑ₖ A[i][k] × B[k][j]Each element of the product matrix is the dot product of a row from A and a column from B.
Where:
A[i][k]= Element in row i, column k of matrix AB[k][j]= Element in row k, column j of matrix BC[i][j]= Resulting element in row i, column jInputs
Result
For the 2×2 matrix [[3,7],[1,5]], the determinant is (3)(5) − (7)(1) = 15 − 7 = 8. Since det ≠ 0, the matrix is invertible.
Inputs
Result
det = 4×6 − 7×2 = 24 − 14 = 10. The inverse is (1/10) × [[6,−7],[−2,4]] = [[0.6,−0.7],[−0.2,0.4]].
Inputs
Result
Each element of the result is the dot product of a row of A and a column of B. For example, C[1][0] = 3×5 + 4×7 = 15 + 28 = 43.
For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc. Multiply the main diagonal (top-left to bottom-right) and subtract the product of the anti-diagonal. If the determinant is zero, the matrix is singular and has no inverse.
| Matrix | Determinant | Invertible? |
|---|---|---|
| [[3,7],[1,5]] | 8 | Yes |
| [[2,4],[1,2]] | 0 | No |
| [[1,0],[0,1]] | 1 | Yes (Identity) |
| [[4,7],[2,6]] | 10 | Yes |
For a 2×2 matrix [[a,b],[c,d]], the inverse is (1/det) × [[d,−b],[−c,a]], where det = ad − bc. Swap the main diagonal elements, negate the off-diagonal elements, and divide by the determinant. The inverse only exists when det ≠ 0.
| Step | 2×2 Method | 3×3 Method |
|---|---|---|
| Determinant | ad − bc | Cofactor expansion |
| Adjugate | Swap & negate | Transpose of cofactor matrix |
| Inverse | (1/det) × adj | (1/det) × adj |
| Complexity | 4 operations | 40+ operations |
Matrix multiplication computes each element C[i][j] by taking the dot product of row i from matrix A and column j from matrix B. For two n×n matrices, each element requires n multiplications and n−1 additions. Matrix multiplication is not commutative: A×B ≠ B×A in general.
The transpose of a matrix swaps its rows and columns: element A[i][j] becomes Aᵀ[j][i]. For a 2×3 matrix, the transpose is 3×2. The transpose of a symmetric matrix equals the original matrix. The determinant of a matrix equals the determinant of its transpose.
A zero determinant means the matrix is singular: it has no inverse, its rows (or columns) are linearly dependent, and the associated linear system has either no solution or infinitely many solutions. Geometrically, the transformation collapses space into a lower dimension.
Last Updated: Mar 11, 2026
This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.