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.
The determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc. For [[3,7],[1,5]], det = 15 − 7 = 8. This single number encodes whether the matrix is invertible (det ≠ 0), how much it scales area (|det| = scale factor), and whether it reverses orientation (det < 0 means a mirror reflection).
For 3×3 matrices, the determinant uses cofactor expansion along the first row: det = a(ei−fh) − b(di−fg) + c(dh−eg). This requires 12 multiplications and 5 additions versus 2 multiplications and 1 subtraction for 2×2. The computational cost grows factorially with naive expansion — an n×n determinant by cofactors requires n! terms, making LU decomposition (O(n³)) essential for matrices larger than 4×4.
A zero determinant signals that rows or columns are linearly dependent. The matrix [[2,4],[1,2]] has det = 0 because row 1 = 2 × row 2. Geometrically, the transformation collapses 2D space onto a line. The Quadratic Equation Calculator encounters determinants when solving systems of two equations in two unknowns via Cramer’s rule.
| Matrix | Determinant | Invertible | Geometric Meaning |
|---|---|---|---|
| [[1,0],[0,1]] | 1 | Yes | Identity — no change |
| [[2,0],[0,3]] | 6 | Yes | Scales area by 6× |
| [[0,1],[-1,0]] | 1 | Yes | 90° rotation |
| [[2,4],[1,2]] | 0 | No | Collapses to a line |
The inverse of a 2×2 matrix [[a,b],[c,d]] is (1/det) × [[d,−b],[−c,a]]. For [[4,7],[2,6]]: det = 24 − 14 = 10, so the inverse is (1/10) × [[6,−7],[−2,4]] = [[0.6,−0.7],[−0.2,0.4]]. Multiplying the original by its inverse produces the identity matrix [[1,0],[0,1]], confirming correctness.
For 3×3 matrices, the process requires computing the cofactor matrix (9 determinants of 2×2 submatrices), transposing it to get the adjugate, and dividing by the determinant. This involves approximately 40 arithmetic operations for a single 3×3 inverse. In practice, Gaussian elimination with partial pivoting is preferred for numerical stability.
Matrix inversion solves linear systems Ax = b directly: x = A⁻¹b. A system of 3 equations in 3 unknowns becomes a single matrix multiplication once A⁻¹ is known. The Standard Deviation Calculator uses matrix operations internally when computing covariance matrices for multivariate data sets.
| Step | 2×2 Method | 3×3 Method | n×n (Gaussian) |
|---|---|---|---|
| Determinant | ad − bc | Cofactor expansion | LU decomposition |
| Adjugate | Swap & negate | 9 cofactor determinants | Row reduction |
| Divide by det | 4 divisions | 9 divisions | n² divisions |
| Operations | ~6 | ~40 | O(n³) |
Matrix multiplication computes C[i][j] as the dot product of row i from A and column j from B. For 2×2 matrices, this requires 8 multiplications and 4 additions. For 3×3: 27 multiplications and 18 additions. The classic algorithm runs in O(n³) time, though Strassen’s algorithm achieves O(n²·⁸ⁱ) by trading multiplications for additions.
A critical property: matrix multiplication is NOT commutative. AB ≠ BA in general. For transformations, this means the order matters. In computer graphics, rotating then translating an object produces a different result than translating then rotating. The matrices [[1,2],[3,4]] × [[5,6],[7,8]] = [[19,22],[43,50]], but reversing the order gives [[23,34],[31,46]].
Every linear transformation in 2D or 3D can be represented as matrix multiplication. Rotation by angle θ: [[cosθ, −sinθ],[sinθ, cosθ]]. Scaling by sx, sy: [[sx,0],[0,sy]]. Reflection across x-axis: [[1,0],[0,−1]]. Computer graphics engines combine dozens of these into a single matrix, then apply it to millions of vertices per frame at 60+ fps. The Trigonometry Calculator computes the sin and cos values needed for rotation matrices.
Matrix multiplication C = AB requires columns(A) = rows(B). The result has dimensions rows(A) × columns(B). A 2×3 matrix times a 3×4 matrix produces a 2×4 result.
Last Updated: Mar 26, 2026
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