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MatrixNotes-1

# MatrixNotes-1 - 1 Operations on Matrices Definition A...

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1 Operations on Matrices Definition: A matrix is a rectangular array of numbers arranged in rows and columns. An m x n matrix is a matrix with m rows and n columns and is usually denoted by A = 11 12 1n 21 22 2n m1 m2 mn a a a a a a a a a = ij a , I = 1,2 …m, j = 1,2 … n The entries could be complex numbers. The matrices 1 3 -1 0 2 4 and 1 0 1 4 2 -3 are 3x2 and 2x3 matrices.` A can be written in the form ij a where i = 1, 2 … m and j = 1, 2 … n A square matrix has the same number of rows and columns. A column matrix has only one column. A zero matrix has zeros for all its entries and is denoted by O. O = 0 0 0 0 An identity matrix has 1 in the main diagonal and zeros everywhere else and is denoted by I . I = 1 0 0 1 or I = 1 0 0 0 1 0 0 0 1

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2 Algebra of Matrices Equality of Matrices Two matrices A and B are equal, written A=B, if they have the same dimensions and their corresponding entries are equal. Sum of Matrices Let A and B be matrices with the same dimensions. The sum of A and B, written A+B, is the matrix obtained by adding corresponding entries of A and B. Multiplication by a Scalar Let A be a matrix and c be a scalar. The scalar multiple cA is the matrix obtained by multiplying each entry of A by the scalar c. Examples 1. Let A = 1 1 2 5 3 1 3A = 3 -3  2. Let A = 1 3 2 4 , B = 0 -1 2 3 A+B = 1 2 4 7 Matrix multiplication Let A and B be matrices such that the number of columns of A match the number of rows of B. Suppose A is m x n and B is n x p matrix, the product AB is the m x p matrix C, defined by C ij = n ik kj k=1 a b In other words, to obtain the entry in the i -th row and j -th column, we take the dot product of the i -th row of A and the j -th column of B. Examples Let A = 1 2 3 -3 -2 -1 B = -4 5 0 4 -5 0 AB = -19 13 17 -23 Transpose of a Matrix
3 The transpose of the m x n matrix A, written A T or A , is the n x m matrix B whose j -th column is j -th row of A. Matrix Form of a Linear System A system of m linear equations in the n unknowns x 1 , x 2 … x n can be written in the general form: a 11 x 1 + 12 2 a x + a 1n x n = b 1 a 21 x 1 + 22 2 a x + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + … a mn x n = b m This system can be written in the following matrix form: 11 12 1n 21 22 2n m1 m2 mn a a - a a a - a - - - a a - a 1 2 - n x x x = 1 2 m b b - b AX = B Where A is a m x n matrix, X is a n x 1 matrix, and B is a m x 1 matrix.

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MatrixNotes-1 - 1 Operations on Matrices Definition A...

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