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Explore matrix organization, dimensions, operations, and solutions for systems of equations, with examples like Gaussian Elimination. Learn about augmented matrices and elementary row operations. Practice solving equations using row echelon form and reduced row-echelon form.
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Let’s do another one: we’ll keep this one Now we’ll use the 2 equations we have with y and z to eliminate the y’s.
multiply the first equation by 8 and add the equations to eliminate y.
Matrix: rectangular organization of real numbers in rows and columns Column 1 C 2 C 3 C n Row 1 Row 2 Row 3 Row m • Entry aijis a real number in a certain “i” row and “j” column
Dimension, or ORDER, of a matrix • Dimension or order is given by m x n m = number of rows n = number of columns
Square Matrix • In a square matrix: m = n; rows = columns • Note: To solve a system of equations, you will need square coefficient matrices
Coefficient Matrix • A matrix whose real entries are the coefficients from a system of equations
Solution Matrix • A column matrix whose entries are the solutions of the system of equations
Augmented matrix- A matrix set up where the coefficient matrix and solution matrix of a system of equations is combinedNotation: Matrix square brackets with a bar separating the coefficients
Elementary Row Operations- Manipulating to Solve Matrices/ Systems 1.) Interchange two rows 2.) Multiply a row by a NONZERO multiple 3.) Add a multiplied row to another row
Row- Echelon Form (ref) Page 548: 1.) If there is a row of all zeros, it is the last row 2.) The first nonzero entry of a nonzero row is 1 3.) The 1’s occur at a diagonal Reduced Row-echelon Form (rref) If every column that has a leading 1 has zeros above and below the 1.
Gaussian Elimination 1.) Write the augmented matrix of a system of equations 2.) Use elementary row operations to rewrite the augmented matrix in ref form 3.) Rewrite the matrix as a system of equations and back-substitute in to solve for each variable
Example 1: Solve Row 2: Added R1 + R2 -4 3 5 -1 3 -1 -3 0 -1 2 2 Row 3: Added -2(R1) + R3 -2 8 -6 -10 2 0 -4 6 0 8 -10 -4
Row 2: -1(R2) 0(-1) (-1)(-1) 2(-1) 2(-1) Row 3: Added 8(R2) + R3 0 -8 16 16 08 -10 -4 0 0 6 12 Row 3: 1/6(R3)