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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.)
In this case, the solution is given by Cramer's rule: = () =,,, …, where is the matrix formed by replacing the -th column of by the column vector . This follows immediately by column expansion of the determinant, i.e.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The unimodular matrix used (possibly implicitly) in lattice reduction and in the Hermite normal form of matrices. The Kronecker product of two unimodular matrices is also unimodular. This follows since det ( A ⊗ B ) = ( det A ) q ( det B ) p , {\displaystyle \det(A\otimes B)=(\det A)^{q}(\det B)^{p},} where p and q are the dimensions of A and ...
Coefficient matrices are used in algorithms such as Gaussian elimination and Cramer's rule to find solutions to the system. The leading entry (sometimes leading coefficient [ citation needed ] ) of a row in a matrix is the first nonzero entry in that row.
The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space. For example, consider the matrix
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.