1. Matrices and Matrix Operations

2. Matrices An m×n matrix A is a rectangular array of mn real numbers arranged in m horizontal rows and n vertical columns. The matrix A has size m×n. The element in the row i and column j of the matrix A is written as a ij. 3 7

3. Matrix Notation a horizontal set of elements is called a row a vertical set is called a column first subscript refers to the row number second subscript refers to column number

4. Matrix Notation row 2 column 3 This matrix has m rows and n columns It has the dimensions m by n (m × n)

5. Types of Matrices Main diagonal

6. Matrix Addition

7. Matrix Scalar Multiplication

8. Matrix Multiplication

10. Two matrices A and B can be multiplied only if the number of columns of A equals the number of rows of B. The element in the i th row and j th column of the product matrix AB is obtained by adding up the product of corresponding elements of the i th row of A and j th column of B.

11. Matrix Multiplication

14. Transpose of a Matrix

15. Symmetric and Skew Symmetric Matrices

17.  An n×n matrix A is said to be invertible (or has inverse) if there exists an n×n matrix B such that AB = BA = I n. The matrix B, denoted by A -1, is called the inverse of A. If there is no such matrix B, then A is called singular, otherwise A is called nonsingular.  Note: The inverse of a matrix is unique. If A and B are nonsingular matrices, then o A -1 is nonsingular and (A -1 ) -1 = A. o A T is nonsingular and (A T ) -1 = (A -1 ) T. o AB is nonsingular and (AB) -1 = B -1 A -1. o A is nonsingular and (A n ) -1 = (A -1 ) n. Inverse of a Matrix

18.  To compute the inverse of an n×n matrix A : Form the n×2n matrix [A | I n ]. Transform this matrix to the reduced row echelon form. There are two possibilities: o Obtaining [I n | A -1 ], the job has been done. o Obtaining [C ≠ I n | ??], C has a zero row, A is singular. Inverse of a Matrix

22.  If AX = B is a linear system of n equations in n unknowns and if A is nonsingular (A is row equivalent to I n ), then the linear system has a unique solution obtained by X = A -1 B.  If A is an n×n singular matrix, then the linear system AX = O has a nontrivial solution (infinitely many solutions). If A is nonsingular, then the homogeneous AX = O has only the trivial solution X = O. Matrix Equations

23. Check that this is the inverse of A

24. Matrix Equations