system of linear equation, Exercises of Mathematics

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SYSTEMS OF LINEAR

EQUATION

AGENDA

 MATRICES o TYPES AND OPERATION o DETERMINANT o INVERSE  SYSTEMS OF LINEAR EQUATION IN MATRIX FORM  DIRECT METHOD AND ITERATIVE METHODS o DIAGONALLY DOMINANT MATRIX o ILL-CONDITIONED MATRIX

MATRIX SPECIAL FORMS  A symmetric matrix is one where the rows equal the columns—that is, 𝑎𝑖𝑗 = 𝑎𝑗𝑖 for all i’s and j’s.  A diagonal matrix is a square matrix where all elements off the main diagonal are equal to zero, as in  An identity matrix is a diagonal matrix where all elements on the main diagonal are equal to 1, as in The identity matrix has properties similar to unity. That is,

MATRIX SPECIAL FORMS  An upper triangular matrix is one where all the elements below the main diagonal are zero, as in.  A lower triangular matrix is one where all elements above the main diagonal are zero, as in  A banded matrix has all elements equal to zero, with the exception of a band centered on the main diagonal: The preceding matrix has a bandwidth of 3 and is given a special name—the tridiagonal matrix.

MATRIX OPERATIONS Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are computed as

MATRIX OPERATIONS Similarly, the subtraction of two matrices, say, [E] minus [F], is obtained by subtracting corresponding terms, as in

MATRIX OPERATIONS The multiplication of a matrix [A] by a scalar g is obtained by multiplying every element of [A] by g.

MATRIX OPERATIONS The product of two matrices is represented as [C] = [A][B], where the elements of [C] are defined as

SAMPLE:

1. [B]+[E]

SAMPLE:

2. [B]-[E]

SAMPLE:

4. [B][E]

SAMPLE:

5. (4)[B][E]

SAMPLE:

6. Identify the Determinant of the Matrix

MATRIX: INVERSE Although multiplication is possible, matrix division is not a defined operation. However, if a matrix [A] is square and nonsingular, there is another matrix [𝐴] − 1 , called the inverse of [A], for which Thus, the multiplication of a matrix by the inverse is analogous to division, in the sense that a number divided by itself is equal to 1. That is, multiplication of a matrix by its inverse leads to the identity matrix. The inverse of a matrix can be represented simply by [A] − 1 =