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CHAPTER 2 MATRICES 2.1 Operations with Matrices Matrix

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1 CHAPTER 2 MATRICES 2.1 Operations with Matrices Matrix
(i, j)-th entry: row: m column: n size: m×n

2 i-th row vector row matrix j-th column vector column matrix Square matrix: m = n

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7 Matrix form of a system of linear equations:
= A x b

8 Partitioned matrices:
submatrix

9 linear combination of column vectors of A
a linear combination of the column vectors of matrix A: = = = linear combination of column vectors of A

10 2.2 Properties of Matrix Operations
Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix: Identity matrix of order n:

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14 (Commutative law for multiplication)
Real number: ab = ba (Commutative law for multiplication) Matrix: Three situations: (Sizes are not the same) (Sizes are the same, but matrices are not equal)

15 Real number: (Cancellation law) Matrix: (1) If C is invertible, then A = B (Cancellation is not valid)

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17 Transpose of a matrix:

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20 A square matrix A is symmetric if A = AT
Symmetric matrix: A square matrix A is symmetric if A = AT Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A Note: is symmetric Proof:

21 2.3 The Inverse of a Matrix Notes:

22 If A can’t be row reduced to I, then A is singular.

23 Power of a square matrix:

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25 Note:

26 Note: If C is not invertible, then cancellation is not valid.

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28 2.4 Elementary Matrices Note:
Only do a single elementary row operation.

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32 Note: If A is invertible

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35 Note: If a square matrix A can be row reduced to an upper triangular matrix U using only the row operation of adding a multiple of one row to another, then it is easy to find an LU-factorization of A.

36 Solving Ax=b with an LU-factorization of A
Two steps: (1) Write y = Ux and solve Ly = b for y (2) Solve Ux = y for x


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