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Transcript

Introduction to Matrices HONORS ADVANCED ALGEBRA II/TRIGONOMETRY MS. LEE Essential Stuff Essential Question: What is a matrix, and how do we perform mathematical operations on matrices? Essential Vocabulary: Matrix Scalar Determinant Inverse What is a matrix? A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns enclosed in a single set of brackets. A= 𝑎1,1 𝑎2,1 𝑎1,2 𝑎2,2 The dimensions of a matrix are the number of horizontal rows and the number of vertical columns it has. NOTE!!: The number of rows always comes before the number of columns. Matrix Terminology Each number, expression, or symbol in a matrix is called an element or an entry. B= 4 −7 11 0 3 6 4ЄB Entries are denoted by a variable and two subscripts (rows and columns). b1,2 = -7 b3,2 = 6 b2,1= 4= Adding and Subtracting Matrices You can add or subtract matrices if and only if they have the same dimensions. In order to add or subtract two or more matrices, add their corresponding elements. Adding and Subtracting Matrices A + B = A+B 𝑒 𝑏 + 𝑔 𝑑 𝑎 𝑐 A 𝑎 𝑐 - 𝑒 𝑏 − 𝑔 𝑑 𝑓 𝑎+𝑒 = ℎ 𝑐+𝑔 B = 𝑏+𝑓 𝑑+ℎ A-B 𝑓 𝑎−𝑒 = ℎ 𝑐−𝑔 𝑏−𝑓 𝑑−ℎ Examples Scalar Multiplication You can multiply a matrix by a constant called a scalar. In order to perform scalar multiplication on a matrix, multiply each element in the matrix by the scalar. Examples Homework Homework 2.1 Matrix Multiplication Matrix multiplication has no operational counterpart in the real number system. In order to multiply two matrices (matrix A and matrix B), the number of columns in A must be equal to the number of rows in matrix B. Matrix A Matrix B Matrix AB 3x2 2x4 3x4 equal Dimensions of AB Matrix Multiplication When multiplying two matrices, A and B, multiply the entries of the first row of matrix A and the first column of matrix B, then add those products up to make the first entry in matrix AB. Repeat this step until we have multiplied each row in matrix A with each column in matrix B. Examples Determinant The determinant is a real number associated with SQUARE matrices. It tells us special things about the matrix useful in solving systems of equations, calculus, and more. Notation: det(A) = |A| 𝑎 𝑐 𝑏 𝑑 , then det(A) = |A|= 𝒂 𝒄 𝒃 𝒅 Let A = = ad – bc. For any matrix larger than 2x2, the determinant will be found using calculator. Examples Inverse of Matrices: Not every matrix has an inverse. A matrix has an inverse if and only if it's determinant is not 0. A matrix is invertible if and only if ad – bc ≠ 0. 𝑎 Let A = 𝑐 𝑏 be invertible. 𝑑 Then, 1 𝑎𝑑−𝑏𝑐 𝐴−1 = 𝑑 −𝑐 −𝑏 and 𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼 𝑎 Homework Homework 2.2