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

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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

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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....
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## 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.

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

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