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Solving Systems Using Matrices

Solving Systems Using Matrices. Inverse Matrices. Preview. Standards and Objectives Defining a Matrix Writing Systems as Matrices Solving a System by the Matrix Equation Why This New Method? Practice. Standards and Objectives. Defining a Matrix.

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Solving Systems Using Matrices

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  1. Solving Systems Using Matrices Inverse Matrices

  2. Preview • Standards and Objectives • Defining a Matrix • Writing Systems as Matrices • Solving a System by the Matrix Equation • Why This New Method? • Practice

  3. Standards and Objectives

  4. Defining a Matrix • A matrix is an array or ordered set of numbers • Each matrix has a name, given by a capital letter such as A • A matrix is “classified” by its number of rows and columns…in that order • Each number in a matrix has an “address”

  5. Example • is a 2x3, read “2 by 3”, matrix named A • Each number is addressed by a lowercase a followed by its row and column • a21 is the number that is in row 2, column 1

  6. Matrices and Systems • Matrix A: the coefficients from the system • Matrix X: 1 column matrix with first variable on top, going down • Matrix B: 1 column matrix with the constants on the right side of the equal sign

  7. Write the matrices for the systems

  8. A word of warning • Notice the x’s and y’s aren’t on the same side • Each system must be in “standard form” of Ax + By = C • Rewrite the system before writing the matrices

  9. The Matrix Equation:AX=B A X = B A-1A X = B A-1 X = B A-1 • Matrix A times Matrix X equals Matrix B • To solve for matrix X, we use the “inverse matrix” A-1 • We will use the calculator to do the calculation part

  10. Write out the other examples using the matrix equation

  11. Entering the Matrix Enter Matrix A Enter Matrix B Select Mat B and press right arrow on D-Pad Give dimensions of Matrix 2 ENTER, 1 ENTER Input constants into matrix 1 ENTER, -3 ENTER • MENU • MAT for Matrix • Select Mat A and press right arrow on D-Pad • Give dimensions of Matrix • 2 ENTER, 2 ENTER • Input coefficients into matrix • 5 ENTER, 3 ENTER, 3 ENTER, 2 ENTER • Press Exit

  12. Solving on the Calculator • Go to RUN • Press OPTN button (next to shift) • F2 for MAT (matrix) • F1- MAT again (puts a Mat on the screen) • ALPHA A • SHIFT x-1 (this gives us the inverse of A) • F1- MAT again • ALPHA B • Press EXE

  13. Reading the Solution • The matrix it gives you as the answer is the x and y values of the system • If you get a Ma Error (Math Error) • Could be no solution • Could be infinite solutions • You will have to solve by hand to figure out which is which

  14. Try Solving the Other Examples

  15. Why this new method? • Tomorrow we will do this all again with systems that have 3 variables. • Imagine doing substitution and elimination with 3 or more different equations. • It is possible, but it takes some time… • For now, let’s practice 2 variable systems

  16. Practice • Write each system as a matrix and use the matrix equation • A X = B • Show the steps: • X = B A-1 • Use the calculator to solve the matrix equation • p.146 #’s 28-30, 37-42 • p.147 #50 & 52

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