DETERMINANT MATRIX YULVI ZAIKA

DETERMINANT a “determinant” is a certain kind of function that associates a real number with a square matrix We will obtain a formula for the inverse of an invertible matrix as well as a formula for the solution to certain systems of linear equations in terms of determinants. is invertible if ad-bc 0 . The expression ad – bc occurs so frequently in mathematics that it has a name; it is called the determinant of the matrix A and is denoted by the symbol det A or |A| . With this notation, the formula A-1for given in

Finding Minors and Cofactors

The definition 3x3 of a determinant in terms of minors and cofactors is

Determinant matrix 3x3

Cofactor Expansion Along the First Column

Adjoint of matrix If A is any nxn matrix and Cij is the cofactor of aij, then the matrix is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj A.

EX If is A an invertible matrix, then

Determinant of an Upper Triangular Matrix If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; that is, det (A)= a11. a22a33…ann.

Cramer's Rule If Ax=b is a system of linear equations in unknowns such that det (A) 0 , then the system has a unique solution. This solution is Where Aj is the matrix obtained by replacing the entries in the j th column of A by the entries in the matrix

Use Cramer's rule to solve

EVALUATING DETERMINANTS BY ROW REDUCTION Let A be a square matrix. If A has a row of zeros or a column of zeros, then det A=0 . Let A be a square matrix. Then det A= det AT . Let A be an nxn matrix (a) If B is the matrix that results when a single row or single column of A is multiplied by a scalar , then det (A)=k det(B) . (b) If B is the matrix that results when two rows or two columns of A are interchanged, then det (B)=-det(A) If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column is added to another column, then det(B)=det(A) .

Elementary Matrices Let E be an nxn elementary matrix. If E results from multiplying a row of In by k, then det E=k. If E results from interchanging two rows of In , then det(E)=-1 . If E results from adding a multiple of one row of to another, then det E=1.

Matrices with Proportional Rows or Columns If A is a square matrix with two proportional rows or two proportional columns, then det (A)=0 .

Evaluating Determinants by Row Reduction

This determinant could be computed as above by using elementary row operations to reduce A to row-echelon form, but we can put A in lower triangular form in one step by adding-3 times the first column to the fourth to obtain

Row Operations and Cofactor Expansion By adding suitable multiples of the second row to the remaining rows, we obtain

PROPERTIES OF THE DETERMINANT FUNCTION

Linear Systems of the Form Ax=x Many applications of linear algebra are concerned with systems of n linear equations in n unknowns that are expressed in the form Ax=x where is a scalar. Such systems are really homogeneous linear systems in disguise, since the equation can be rewritten as x-Ax=0 or, by inserting an identity matrix and factoring, as (I-A)x=0

Finding I-A  Is called a characteristic value or an eigenvalue* of A and the nontrivial solutions of eq are called the eigenvectors of A corresponding to .

The factored form of this equation is (+2)(-5), so the eigenvalues of A are -2 and 5. Jika =5 Jika =-2