TOPICS MATRIX, VECTOR AND MATRIX ALGEBRA SCALAR, VECTOR AND MATRIX A scalar is a number. Examples of scalars are temperature, distance, speed, or mass – all quantities that have a magnitude but no “direction”, other than perhaps positive or negative. Matrix is a rectangular collection of variables or scalars contained with a set of square [ ] or round ( ) brackets. A matrix is consist of m rows and n columns. SCALAR, VECTOR AND MATRIX A vector is a list of numbers. There are (at least) two ways to interpret what this list of numbers mean: One way to think of the vector as being a point in a space. Then this list of numbers is a way of identifying that point in space, where each number represents the vector’s component that dimension. Another way to think of a vector is a magnitude and a direction, e.g. a quantity like velocity (“the fighter jet’s velocity is 250 mph north-by-northwest”). In this way of think of it, a vector is a directed arrow pointing from the origin to the end point given by the list of numbers. SCALAR, VECTOR AND MATRIX Matrix is a rectangular collection of variables or scalars contained with a set of square [ ] or round ( ) brackets. A matrix is consist of m rows and n columns. IN SIMPLE ...... CLASSIFICATION OF MATRIX SQUARE MATRIX DIAGONAL MATRIX CLASSIFICATION OF MATRIX INDENTITY MATRIX SCALAR MATRIX EQUALITY OF MATRIX TWO MATRICES ARE EQUAL IF THEY HAVE THE SAME NUMBER OF ROWS AND COLUMNS AND THEIR CORRESPONDING ENTRIES ARE ALSO EQUAL. ADDITION AND SUBTRACTION OF MATRIX Addition (or subtraction) of two matrices can be accomplished by adding ( or subtracting) the corresponding entries of two matrices which have the same shape. MULTIPLICATION OF MATRIX Multiplication of matrix can be done only if the numer of columnc of the left-hand is matrix is equal to the number of rows of the right-hand matrix. Multiplication is accomplihed by multiplying the elements in each right-hand matrix column, adding their products, then placing the sum at the intersection point of the involved row and column DIVISION OF MATRIX Division of matrices can be accomplished only by multiplying the inverse of the denominator matrix. THE DETERMINANT OF A MATRIX The determinant D, is a scalar calculated from a square matrix. The determinant of a matrix is indicated by enclosing the matrix by vertical lines. PROPERTIES OF DETERMINANTS A. If a matrix has a row or column of zeros, the determinant is zero. B. If a matrix has two identical rows or columns, the determinant is zero. C. If a matrix is triangular, the determininant is equal to the product of the diagonal entries PROPERTIES OF DETERMINANTS C. If a matrix is triangular, the determininant is equal to the product of the diagonal entries D. The value of the determinant is not changed if corresponding rows and columns are changed PROPERTIES OF DETERMINANTS E. if each of a column or row of a determinant is multiplied by m, the value of the determinant is multiplied by m. F. If teo columns or rows of a deteminant are interchanged the sign is changed PROPERTIES OF DETERMINANTS G. The value of a determinant is not changed if each element of a column or row is multiplied by a number k and added or subtracted to the corresponding elements of a column or row. MINORS AND CO FACTORS CALCULATING DETERMINANTS example