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Section 7.2: matrix Algebra

Section 7.2: matrix Algebra. May 16, 2014. Order and elements. The order of a matrix is described as “rows x columns.” Each value in a matrix is given a name, based on its location. For example, the value in the 2 nd row, 3 rd column is . Example: Given ,

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Section 7.2: matrix Algebra

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  1. Section 7.2: matrix Algebra May 16, 2014

  2. Order and elements • The order of a matrix is described as “rows x columns.” • Each value in a matrix is given a name, based on its location. For example, the value in the 2nd row, 3rd column is . Example: Given , • What is the order of the matrix? • What is ?

  3. Special Matrices • A square matrix has the same number of rows as columns. (2x2, 3x3, etc.) • An identity matrix is a square matrix with the diagonal (from top left to bottom right) with values of 1, and all other elements with a value of 0.

  4. Addition, Subtraction, and Scalar Multiplication of Matrices • Add or subtract matrices with the same order. Just add/subtract like elements. • Scalar Multiplication: Distribute the scalar(constant) through to each element of the matrix.

  5. Matrix Multiplication • To multiply matrices, the number of columns from the first matrix must match the number of rows of the second matrix. The resulting matrix will have the number of rows of the first matrix, and the number of columns of the second matrix. • To find element , find the sum of the product of the elements in row 1 of the first matrix and column 1 of the second matrix. Continue using this pattern.

  6. Determinant of a Matrix • Find the determinant of a 2 x 2 matrix, , by ad – bc. Notation: . Evaluate: • To find the determinant of a 3 x 3 matrix, expand by minors. Evaluate:

  7. Inverse matrix • To find the inverse of , . Find the inverse of . Find the inverse of .

  8. Non-Calculator Examples • Find x: • Find y:

  9. Cryptography over the years

  10. Cryptography using matrices • In your group, choose a 3 x 3 matrix that has an inverse matrix. Write this matrix (label it with your group letter) on an index card. Write its inverse on a separate card, again labeling it with your group letter. • Now, each group should follow the directions on the Cryptography Sheet. • Each person should choose their own message, and then encode it with the encoding matrix of your group. • Write all of your original messages on an “answer key” card. • Once all groups have encoded messages, the encryption matrices, answer key card, and encoded messages will be handed in to Mrs. Fehling.

  11. Index Cards • Card 1: Encryption Matrix • Card 2: Decryption Matrix (Inverse of Encryption Matrix) • Card 3: Answer Key Card w/ all of your original messages. • Cards 4 – 6 or 7: Your encrypted message via matrix chunks.

  12. Decode a message • Finally, receive an encoded message with an encryption matrix. • As a group, find the inverse of the encryption matrix. • Use this information to decode the message. • Last, check the accuracy of the decoded message against the answer key card.

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