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Unit 3: Matrices
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Matrix: A rectangular arrangement of data into rows and columns, identified by capital letters. Matrix Dimensions: Number of rows, m, by the number of columns, n. Read as “m by n” matrix. Also known as the order of a matrix. ◦ RBC (ROWS BY COLUMNS)
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Determine the dimensions of each matrix.
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Elements Matrix Element: Each number in a matrix, identified by its row and column. Example: a mn Refers to the m-th row and n-th column
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Example Identify each element. 1. a 23 2. a 12 3. a 31 4. a 21
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Adding and Subtracting Matrices When matrices have the same dimension you add and subtract them by adding or subtracting each corresponding element.
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Add or Subtract the following matrices:
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Using Scalar Products
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Matrix Multiplication When multiplying matrices A and B, the number of COLUMNS in matrix A MUST be equal to the ROWS in matrix B. The size of the product is: # rows in A x # columns in B.
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How to multiply matrices Multiply the elements of each row in the first matrix by the elements in each column of the second matrix Add the products to get the new element.
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Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be?? 1. x 2. x
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Matrix Multiplication
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Equivalent Matrices
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DETERMINANT OF MATRICES
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A special number that can be calculated from the matrix. It tells us things about the matrix that are useful in systems of linear equations, in calculus, and more The symbol for determinant is two vertical lines either side Determinant of a Matrix
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Determinant of a 2x2
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Find the determinant of the following 2x2 matrices:
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Determinant of a 3x3 Matrix
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Find the determinant of the following.
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MATRIX EQUATIONS
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Matrix Equation Example
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Solve each equation:
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INVERSE OF MATRICES
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For matrices, there is no such thing as division. You can add, subtract, and multiple matrices, but you cannot divide them. There is a related concept called inversion
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AX=C Using Inverses to Solve For X
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Inverse Notation REMEMBER we denote inverse with a -1 power So the inverse of matrix A is A -1
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Requirement to have an Inverse Matrix MUST be square, meaning it has the same number of rows and columns Matrix MUST NOT have a determinant of zero.
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Inverse exist?! Does the inverse exist?!?!
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Multiplying Inverse When you Multiply a matrix A times it’s inverse, the Product is the Identity Matrix. Identity Matrix is a square matrix where the top left to Bottom right diagonal are all ones, and everything else is a zero
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Determine if the following matrices are inverses. 1. 2.
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Finding the Inverse of a 2x2 IF THEN
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Find the inverse of the following matrix.
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Use your calculator! 1. 2 nd Matrix Edit 2. Put in your matrix 3. 2 nd Matrix NAME 4. Get your matrix 5. X -1
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The inverse of a matrix can be used when solving matrix equations. For Matrices A and B, we can find Matrix X: IF AX = B THEN X = A -1 B
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*Solve for X: X = A -1 B
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You Try! Solve Each Matrix Equation:
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Solutions: Solve each matrix equation.
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