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4.4 & 4.5 Notes

4.4 & 4.5 Notes. Identity Matrices. Remember :. Identity Matrices :. If the product of two matrices equal the identity matrix then they are inverses. IDENTITY MATRIX PROOF. a = (-3)(1) + (4)(0) = -3 b = (-3)(0) + (4)(1) = 4 c = (-2)(1) + (6)(0) = -2 d = (-2)(0) + (6)(1) = 6.

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4.4 & 4.5 Notes

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  1. 4.4 & 4.5 Notes

  2. Identity Matrices Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.

  3. IDENTITY MATRIX PROOF a = (-3)(1) + (4)(0) = -3 b = (-3)(0) + (4)(1) = 4 c = (-2)(1) + (6)(0) = -2 d = (-2)(0) + (6)(1) = 6

  4. The Inverse formula of a 2x2 Matrix Step 1) Find determinant A scalar, Put under 1 Step 2) Switch a & d Step 3) Change the signs of b&c Step 4) Multiply scalar

  5. FIND THE INVERSE Step 1: Find Determinant A (scalar), put under 1

  6. Step 2: SWITCH 5 AND -2 Step 3: Change signs of 3 and 1 Step 4: Multiply scalar Answer:

  7. 1. Find the inverse Steps 2 & 3 Step 1 Steps 4: Multiply scalar Answer =

  8. Solving Matrix Equations • Find the inverse of the matrix next to the variable • Multiply both sides by the inverse matrix, the inverse must be on the left side when multiplying -Check for the Identity matrix

  9. Step 1:Find the Inverse Matrix First

  10. Step 2 (Multiply both sides by the inverse matrix on the left) Multiply rows by columns Multiply rows by columns Solution

  11. Solve the Matrix Equation Find the inverse first!!!! Multiply both sides by the inverse matrix)

  12. Subtract Matrix from both sides Find the inverse Multiply inverse by both sides (keep it left)

  13. Homework: Read section 4.4 ***Define Identity and Inverse Matrices Pgs. 227-229; 1-3, 14-32e, 54-60e

  14. 4.5 Solving systems using matrices.

  15. A system can be written as a single matrix equation. Linear system A X = B Matrix equation • Matrix A is called the Coefficient matrix. • Matrix X is called the Variable matrix • Matrix B is called the Constant matrix

  16. Solving for x and y Step 1: Set up the equation in matrix form Step 2: Find the inverse of the Coefficient Matrix and multiply both sides

  17. Step 2:Finding the Inverse Matrix

  18. Step 2: Multiply both sides by the Inverse… (2,-2)

  19. USE AN INVERSE MATRIX TO SOLVE THE LINEAR SYSTEM. FIRST, BEGIN BY WRITING THE EQUATIONS IN MATRIX FORM. SECOND, YOU MUST NOW FIND THE INVERSE OF THE COEFFICIENT MATRIX.

  20. FIND THE INVERSE OF THE COEFFICIENT MATRIX.

  21. SOLVE THE SYSTEM BY MULTIPLYING BY THE INVERSE x = 1 AND y = 2 OR (1,2)

  22. More Practice 1. 3. 2. 4. (4,4) (8,4) (-1,-5) (44/5, -26/5)

  23. Homework Read section 4.5 ***Matrix of variables and Matrix of constants Pg.233-235; 1-3, 12-18e, 24-30e, 48-62e

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