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Mastering Inverse Matrices for Linear Systems

Learn to find inverse of square matrices, solve matrix equations using inverses, and tackle linear systems efficiently with inverse matrices. Practice exercises included.

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Mastering Inverse Matrices for Linear Systems

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  1. Use Inverse Matrices to Solve Linear Systems Objectives • To find the inverse of a square matrix • To solve a matrix equation using inverses • To solve a linear system using inverse matrices

  2. Warm-Up Imagine you lived in a world that never defined subtraction or division. What would you do to get the same results?

  3. Objective 1 You will be able to find the inverse of a square matrix

  4. Identity Matrix The identity matrix has 1s along the main diagonal and zeros everywhere else.

  5. Identity Matrix Recall that multiplying a number by 1 gives you back the same number. This is the Identity Property of Multiplication. Similarly, multiplying a matrix by the identity matrix, , will return the original matrix.

  6. Exercise 1 Find : and

  7. Inverse Matrix The inverseof a square matrix is another square matrix such that and . The inverse of matrix is denoted as The inverse is only defined for a square matrix with a determinant

  8. Exercise 2 Find . What is the relationship between , , and ? and

  9. Inverse Matrix (2x2)

  10. Exercise 3 Find :

  11. Exercise 4 Find :

  12. Objective 2 You will be able to solve a matrix equation using inverses

  13. Matrix Equations In general a matrix equation can be written as , where , , and are matrices. To solve this equation for , you would ordinarily divide by . However, there is no matrix division. Instead you solve for by multiplying both sides of the equation by the inverse of .

  14. Matrix Equations In general a matrix equation can be written as , where , , and are matrices. How to solve a matrix equation Since matrix multiplication is not commutative, you must remember to always multiply on the same side.

  15. Exercise 5 Solve the matrix equation for .

  16. Exercise 6 Solve the matrix equation for .

  17. Objective 3 You will be able to solve a linear system using inverse matrices

  18. Exercise 7 Use an inverse matrix to solve the linear system.

  19. Solving Systems using Inverses Write the system as a matrix equation Find the inverse of matrix Multiply both left sides of by Step 1 Step 2 Step 3 is the coefficient matrix The solution is is the constant matrix

  20. Protip: Multiplying by Inverse When multiplying by an inverse matrix to solve a matrix equation, first multiply by the “sub-inverse” and then divide each element by the determinant. “Sub-inverse” 1 divided by the determinant

  21. Exercise 8 Solve each system using inverse matrices.

  22. Inverse Matrix (3x3) The inverse of a matrix is

  23. Inverse Matrix (3x3) The inverse of a matrix is just use a calculator!

  24. Exercise 9 Find .

  25. Exercise 10 Find .

  26. Exercise 11 Solve the system using inverse matrices.

  27. Exercise 12 Solve the system using inverse matrices.

  28. 3.8: Use Inverse Matrices to Solve Linear Systems Objectives • To find the inverse of a square matrix • To solve a matrix equation using inverses • To solve a linear system using inverse matrices

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