M1410802 - (Section 8.2: Operations with Matrices) 8.34...

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(Section 8.2: Operations with Matrices) 8.34 SECTION 8.2: OPERATIONS WITH MATRICES We will not discuss augmented matrices until Part G . For now, we will simply think of a matrix as a box of numbers. PART A: NOTATION The matrix A = a ij , meaning that A consists of entries labeled a ij , where i is the row number, and j is the column number. Example If A is 2 × 2 , then A = a 11 a 12 a 21 a 22 . Note: a 12 and a 21 are not necessarily equal. If they are, then we have a symmetric matrix, which is a square matrix that is symmetric about its main diagonal. An example of a symmetric matrix is: 2 3 3 7
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(Section 8.2: Operations with Matrices) 8.35 PART B: WHEN DOES A = B ? Two matrices (say A and B ) are equal They have the same size, and they have the same numbers (or expressions) in the same positions. Example 1 2 3 4 = 1 2 3 4 Example 1 2 3 4 1 3 2 4 If the matrix on the left is A , then the matrix on the right is A T (“ A transpose”). For the two matrices, the rows of one are the columns of the other. Example 0 1 0 1 The two matrices have different sizes. The matrix on the left is 1 × 2 . It may be seen as a row vector, since it consists of only 1 row. The matrix on the right is 2 × 1 . It may be seen as a column vector, since it consists of only 1 column. Observe that the matrices are transposes of each other. Think About It: What kind of matrix is, in fact, equal to its transpose?
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(Section 8.2: Operations with Matrices) 8.36 PART C: BASIC OPERATIONS Matrix addition: If two or more matrices have the same size, then you add them by adding corresponding entries. If the matrices do not have the same size, then the sum is undefined. Matrix subtraction problems can be rewritten as matrix addition problems. Scalar multiplication: To multiply a matrix by a scalar (i.e., a real number in this class ), you multiply each entry of the matrix by the scalar. Example If A = 2 0 1 1 3 2 B = 0 0 0 0 0 0 B is the 2 × 3 zero matrix, denoted by “0” or “ 0 2 × 3 ” – it is the additive identity for the set of 2 × 3 real matrices. However, when we refer to “identity matrices,” we typically refer to multiplicative identities, which we will discuss later. C = 1 2 0 0 1 3 then …
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(Section 8.2: Operations with Matrices) 8.37 1) Find A + B + 2 C A + B + 2 C = 2 0 1 1 3 2 + 0 0 0 0 0 0 Perform matrix addition.   + 2 1 2 0 0 1 3 Perform scalar multiplication   = 2 0 1 1 3 2 + 2 4 0 0 2 6 = 4 4 1 1 5 4 2) Find A 5 C A 5 C = A + 5 ( ) C = 2 0 1 1 3
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410802 - (Section 8.2: Operations with Matrices) 8.34...

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