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**Matrices & Systems of Linear Equations**

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Special Matrices

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Special Matrices

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**corresponding entries are equal**

Equality of Matrices Two matrices are said to be equal if they have the same size and their corresponding entries are equal

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Equality of Matrices Use the given equality to find x, y and z

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**Matrix Addition and Subtraction Example (1)**

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**Matrix Addition and Subtraction Example (2)**

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**Multiplication of a Matrix by a Scalar**

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**The result is a (n by k) Matrix**

Matrix Multiplication (n by m) Matrix X (m by k) Matrix The 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

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**Matrix Multiplication 3x3 X 3x3**

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**Matrix Multiplication 1x3 X 3x3→ 1x3**

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Example (1)

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Example (2) (1X3) X (3X3) → 1X3

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Example (3) (3X1) X (1X2) → 3X2

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Example (4)

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Transpose of Matrix

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**Properties of the Transpose**

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**Matrix Reduction Definitions (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.

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**Matrix Reduction Definitions (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.

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**Examples of Reduced Matrices**

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**Examples matrices that are not reduced**

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**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.

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Interchanging Rows

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**Replacing a row by a nonzero multiple of itself**

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**Replacing a row by the sum of that row and a nonzero multiple of another row**

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**Augmented Matrix Representing a System of linear Equations**

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**Solving a System of Linear Equations by Reducing its Augmented Matrix Using Row Operations**

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Solution

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Solution of the System

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**The Idea behind the Reduction Method**

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**Interchanging the First & the Second Row**

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**Multiplying the first Equation by 1/3**

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**Subtracting from the Third Equation 5 times the First Equation**

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**Subtracting from the First Equation 2 times the Second Equation**

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**Adding to the Third Equation 12 times the Second Equation**

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**Dividing the Third Equation by 40**

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**Adding to the First Equation 7 times the third Equation**

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**Subtracting from the Second Equation 3/2 times the third Equation**

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**Systems with infinitely many Solutions**

x=3-2r y = r 3 5 -1 1 -17 10

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**Systems with infinitely many Solutions**

y=-r x=-3r z=r -1 -3 1 10 30 -10 -1/3 1/3

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Details of reduction

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**Systems with no Solution**

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**Details of the reduction**

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**Finding the Inverse of an nXn square Matrix A**

1. Adjoin the In identity matrix to obtain the Augmented matrix [A| In ] 2. Reduce [A| In ] to [In | B ] if possible Then B = A-1

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Example (1)

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Example (2)

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**Inverse Matrix The formula for the inverse of a 2X2 Matrix**

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**Using the Inverse Matrix to Solve System of Linear Equations**

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Problem

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Homework

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