# determinant of a+b

Determinants are of use in ascertaining whether a system of n equations in n unknowns has a solution. Let σ \sigma σ be a permutation of {1, 2, 3, …, n} \{1, 2, 3, \ldots, n\} {1, 2, 3, …, n}, and S S S the set of those permutations. Evaluating large determinants can be tedious and we will use computers wherever possible (see box at right). by M. Bourne. Large Determinants. To calculate a determinant you need to do the following steps. If A is an elementary matrix and B is an arbitrary matrix of the same size then det(AB)=det(A)det(B). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … An alternate method, determinant by permutations, calculates the determinant using permutations of the matrix's elements. If B is an n × 1 vector and the determinant of A is nonzero, the system of equations AX = B always has a solution. Then the determinant of an n × n n \times n n × n matrix A A A is Expanding 4×4 Determinants For a 2×2 Matrix. But if you have to do large determinants on paper, here's how.. rows are switched, the matrix is unchanged, but the determinant is negated. The determinant of a product AB is the product of the determinants of A and B. 2. Multiply the main diagonal elements of the matrix - determinant is calculated. 2 Corollary 6 If B is obtained from A by adding ﬁ times row i to row j (where i 6= j), then det(B) = det(A). A is obtained from I by adding a row multiplied by a number to another row. Set the matrix (must be square). (This is a row operation of type 3.) The determinant of a matrix can be found using cofactor expansion along any row or column. For a 2×2 matrix (2 rows and 2 columns): [source: mathisfun] The determinant is: |A| = ad − bc or t he determinant of A equals a × d minus b × c. It is easy to remember when you think of a cross, where blue is positive that goes diagonally from left to right and red is negative that goes diagonally from right to left. The next two properties follow from this. Reduce this matrix to row echelon form using elementary row operations so that all the elements below diagonal are zero. = a 1 (b 2 c 3 – b 3 c 2) – a 2 (b 1 c 3 – b 3 c 1) + a 3 (b 1 c 2 – b 2 c 1) By interchanging the rows and columns of Δ, we get the determinant Expanding Δ 1 along first column, we get, Proof: Let C be the matrix obtained from A by replacing row j with row i. In this case by the first theorem about elementary matrices the matrix AB is obtained from B by adding one row multiplied by a number to another row. A matrix and its transpose have the same determinant. The determinant is multiplicative: for any square matrices A,B of the same size we have det(AB) = (det(A)) (det(B)) [6.2.4, page 264]. 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