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Matrices & Systems of Linear Equations

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## Matrices & Systems of Linear Equations

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**Equality of Matrices**Two matrices are said to be equal if they have the same size and their corresponding entries are equal**Equality of Matrices**Use the given equality to find x, y and z**Matrix Multiplication(n by m) Matrix X (m by k) MatrixThe**number of columns of the matrix on the left= number of rows of the matrix on the right The result is a (n by k) Matrix**Matrix ReductionDefinitions (1)**1. Zero Row:A row consisting entirely of zeros 2. Nonzero Row:A row having at least one nonzero entry 3. Leading Entry of a row:The first nonzero entry of a row.**Matrix ReductionDefinitions (2)**Reduced Matrix: A matrix satisfying the following: 1. All zero rows, if any, are at the bottom of the matrix 2. The leading entry of a row is 1 3. All other entries in the column in which the leading entry is located are zeros. 4. A leading entry in a row is to the right of a leading entry in any row above it.**Elementary Row Operations**1. Interchanging two rows 2. Replacing a row by a nonzero multiple of itself 3. Replacing a row by the sum of that row and a nonzero multiple of another row.**Replacing a row by the sum of that row and a nonzero**multiple of another row**Solving a System of Linear Equations by Reducing its**Augmented Matrix Using Row Operations**Subtracting from the Third Equation 5 times the First**Equation**Subtracting from the First Equation 2 times the Second**Equation**Subtracting from the Second Equation 3/2 times the third**Equation