DETERMINANTS P.S. Laplace ( - ) DETERMINANTS This may be thought of as a function which associates each square matrix with a unique number (real or complex). If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k , where A ∈ M and k ∈ K, then f (A) is called the determinant of A. It is also denoted by |A| or det A or ∆ .
If A = , then determinant of A is written as |A| = d = det (A) Remarks (i) For matrix A, |A| is read as determinant of A and not modulus of A. (ii) Only square matrices have determinants. . .
Determinant of a matrix of order one Let A = [ a ] be the matrix of order , then determinant of A is defined to be equal to a . . Determinant of a matrix of order two Let A = be a matrix of order × , then the determinant of A is defined as: det (A) = |A| = ∆ = = a a – a a Example Evaluate . Solution We have = ( ) – (– ) = + = .
Example Evaluate – Solution We have – = x ( x ) – ( x + ) ( x – ) = x – ( x – ) = x – x + = . . Determinant of a matrix of order × Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column).
There are six ways of expanding a determinant of order corresponding to each of three rows (R , R and R ) and three columns (C , C and C ) giving the same value as