matrices_algebra

# matrices_algebra - MIT OpenCourseWare http/ocw.mit.edu...

This preview shows pages 1–4. Sign up to view the full content.

MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
M. Matrices and Linear Algebra 1. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices. In general, they need not be square, only rectangular. A rectangular array of numbers having m rows and n columns is called an m x n matrix. The number in the i-th row and j-th column (where 1 5 i 5 m, 1 5 j 5 n) is called the ij-entry, and denoted aij; the matrix itself is denoted by A, or sometimes by (aij). Two matrices of the same size are equal if corresponding entries are equal. Two special kinds of matrices are the row-vectors: the 1 x n matrices (al, az, . . . , a,); and the column vectors: the m x 1 matrices consisting of a column of m numbers. From now on, row-vectors or column-vectors will be indicated by boldface small letters; when writing them by hand, put an arrow over the symbol. Matrix operations There are four basic operations which produce new matrices from old. 1. Scalar multiplication: Multiply each entry by c : cA = (caij) 2. Matrix addition: Add the corresponding entries: A + B = (aij + bij); the two matrices must have the same number of rows and the same number of columns. 3. Transposition: The transpose of the m x n matrix A is the n x m matrix obtained by making the rows of A the columns of the new matrix. Common notations for the transpose are AT and A'; using the first we can write its definition as AT = (aji). If the matrix A is square, you can think of AT as the matrix obtained by flipping A over around its main diagonal. 2 -3 1 5 Example 1.1 Let A = . FindA+B, AT, 2A-3B.
2 18.02 NOTES 4. Matrix multiplication This is the most important operation. Schematically, we have m x n n x p m x P The essential points are: 1. For the multiplication to be defined, A must have as many columns as B has rows; 2. The ij-th entry of the product matrix C is the dot product of the i-th row of A with the j-th column of B. The two most important types of multiplication, for multivariable calculus and differential equations, are: 1. AB, where A and B are two square matrices of the same size - these can always be multiplied; 2. Ab, where A is a square n x n matrix, and b is a column n-vector. Laws and properties of matrix multiplication M-1. A(B + C) = AB + AC, (A + B)C = AC + B C distributive laws M-2. (AB)C = A(BC); (cA) B = c(AB). associative laws In both cases, the matrices must have compatible dimensions. M-3. Let I3 = ; then AI = A and I A = A for any 3 x 3 matrix. I is called the identity matrix of order 3. There is an analogously defined square identity matrix In of any order n, obeying the same multiplication laws. M-4. In general, for two square n x n matrices A and B, AB # BA: matrix multiplication is not commutative. (There are a few important exceptions, but they are very special - for example, the equality A I = IA where I is the identity matrix.) M-5. For two square n x n matrices A and B, we have the determinant law:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern