notes2-1 - of A be the rst column of the new matrix, and so...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
SECTION 2.1 MATRIX ALGEBRA You can multiply matrices by scalars (numbers); when two matrices are the same size you can add and subtract them; when two matrices are appropriate sizes you can multiply them. We know about the matrix-vector product A x , so to understand a matrix-matrix product AB we treat the matrix B column by column. The algebra generated by all this is nice [Theorem 1, page 108; Theorem 2, page 113] EXCEPT that matrix multiplication is not commutative, that is, AB may not equal BA even when both are defined. And you have to be a little careful about sizes. For each positive integer m there is an m × m matrix I m , called an identity matrix , which has 1’s down the main diagonal and 0’s elsewhere and satisfies I m A = A and BI m = B whenever the products are defined. If we interchange the rows and columns of a p × q matrix A , that is, make the first row
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: of A be the rst column of the new matrix, and so on, we get a q p matrix A T , called the transpose of A . Transposes preserve scalar multiplication, addition, and subtraction, but reverse multiplication [Theorem 3, page 115]. What does this mean, exactly???? EXAMPLE. Suppose that A = " 1-2 1 3-4 5 # and B = 1 9-2 3 1 . Compute the following. 1. 2 A + 3 B T 2. AB 3. BA 4. B T A T . HOMEWORK: SECTION 2.1 EXAMPLES, IF TIME. Suppose the third column of matrix B is the sum of the rst two columns of B . What can you say about the third column of AB ? Why? Suppose A and C are matrices such that CA = I n . Show that the equation A x = has only the trivial solution. Explain why A cannot have more columns than rows....
View Full Document

This note was uploaded on 05/07/2010 for the course M 56945 taught by Professor Danielallcock during the Spring '09 term at University of Texas at Austin.

Page1 / 3

notes2-1 - of A be the rst column of the new matrix, and so...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online