Cofactor Expansion
Compute the determinant of a 2x2 matrix using cofactor expansion. Enter a,b,c,d of [[a,b],[c,d]] to get the determinant and cofactor matrix. Free online linear algebra tool with step-by-step results.
About This Calculator
The Cofactor Expansion calculator computes the determinant and cofactor matrix of any 2x2 matrix A = [[a, b], [c, d]] using cofactor expansion along the first row. Cofactor expansion (also known as Laplace expansion) is a fundamental technique in linear algebra for computing determinants that extends to matrices of any size.
For a 2x2 matrix, the determinant is computed as det(A) = a x d - b x c. The cofactor matrix is computed as C = [[+d, -c], [-b, +a]], where each cofactor is the signed minor: C_ij = (-1)^(i+j) x M_ij. For position (1,1), the minor M11 = d and the sign is +, giving C11 = +d. This calculator is ideal for students learning linear algebra, preparing for exams, or verifying manual calculations of determinants and cofactor matrices.
Frequently Asked Questions
What is cofactor expansion for a 2x2 matrix?
Cofactor expansion is a method to compute the determinant of a matrix by expanding along a row or column. For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. The cofactor of a is +d, cofactor of b is -c, and so on. This calculator shows both the determinant and the full cofactor matrix.
How do you calculate the determinant using cofactor expansion?
For a 2x2 matrix [[a, b], [c, d]], the determinant det(A) = a x d - b x c. Using cofactor expansion along the first row: det(A) = a x C11 + b x C12, where C11 = +d and C12 = -c, giving axd + bx(-c) = ad - bc.
What is the sign pattern for cofactor expansion?
The sign for each position follows (-1)^(i+j), forming a checkerboard pattern: position (1,1): +, (1,2): -, (2,1): -, (2,2): +. This sign pattern is applied to each minor (the determinant of the submatrix after removing the row and column) to form the cofactor.
Is the cofactor expansion calculator free to use?
Yes, all calculators on Calculy are completely free to use. No registration, subscriptions, or hidden fees required.
What is the cofactor matrix of a 2x2 matrix?
For a 2x2 matrix A = [[a, b], [c, d]], the cofactor matrix C = [[d, -c], [-b, a]]. Each element C_ij is the cofactor of a_ij. The cofactor matrix is used to find the adjugate (its transpose) and then the inverse of the original matrix.
Can I share my matrix calculation results?
Yes. After entering matrix values and calculating, the values are saved in the page URL. You can bookmark the URL or share it to reproduce the exact calculation.