An Introduction To Linear Algebra by Kenneth Kuttler

By Kenneth Kuttler

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Also if x is a scalar, xA = (cij ) where cij = xaij . The number Aij will typically refer to the ij th entry of the matrix, A. The zero matrix, denoted by 0 will be the matrix consisting of all zeros. Do not be upset by the use of the subscripts, ij. The expression cij = aij + bij is just saying that you add corresponding entries to get the result of summing two matrices as discussed above. Note there are 2 × 3 zero matrices, 3 × 4 zero matrices, etc. In fact for every size there is a zero matrix.

When this has been done, B = A−1 . 20. 25 Let A =  1 −1 1 . Find A−1 . 1 1 −1 Form the augmented matrix,  1  1 1 0 −1 1  1 1 0 0 1 0 1 0 . −1 0 0 1 Now do row operations untill the n × n matrix on the left becomes the identity matrix. This yields after some computations,   1 1 1 0 0 0 2 2  0 1 0 1 −1 0  0 0 1 1 − 21 − 12 and so the inverse of A is the matrix on the right,   1 1 0 2 2  1 −1 0 . 1 1 − 2 − 21 Checking the answer is easy. Just multiply the matrices and see if     1 1 1 0 1 0 1 0 2 2  1 −1 1   1 −1 0 = 0 1 1 1 −1 0 0 1 − 12 − 12 Always check your answer because if mistake.

Amn Bnj         Bnj .  The second entry of this m × 1 matrix is m A2k Bkj . k=1      which is a m × 1 matrix or column vector which equals      A11 A12 A1n  A21   A22   A2n       ..  B1j +  ..  B2j + · · · +  ..  .   .   . 1. MATRICES 39 Similarly, the ith entry of this m × 1 matrix is m Ai1 B1j + Ai2 B2j + · · · + Ain Bnj = Aik Bkj . 3. This motivates the definition for matrix multiplication which identifies the ij th entries of the product. 7 Let A = (Aij ) be an m × n matrix and let B = (Bij ) be an n × p matrix.

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