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Matrix Inverse and Identity Operations Explained

Explore the concepts of matrix inverses and identities with Jeffrey Bivin from Lake Zurich High School. Learn to find inverse matrices and the identity matrix for 2x2 matrices, along with solving systems using inverse matrices. Discover formulas and methods for determining if matrices are inverses of each other.

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Matrix Inverse and Identity Operations Explained

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  1. Inverse & Identity MATRICES Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: October 12, 2005

  2. Inverses and Identities 5x = 3 Jeff Bivin -- LZHS

  3. Now with Matrices ? This is the Identity Matrix for 2 x 2 Matrices Jeff Bivin -- LZHS

  4. Let’s look at another example Jeff Bivin -- LZHS

  5. New Question ? What do we multiply a matrix by to get the Identity? Jeff Bivin -- LZHS

  6. INVERSE The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

  7. INVERSE The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

  8. INVERSE The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

  9. A B This is our Formula! Jeff Bivin -- LZHS

  10. A B This is our Formula! Jeff Bivin -- LZHS

  11. A B C This is our Formula! Jeff Bivin -- LZHS

  12. Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Identity, therefore, INVERSE Matrices Jeff Bivin -- LZHS

  13. Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Not the Identity, therefore, NOT INVERSE Matrices Jeff Bivin -- LZHS

  14. Does the Matrix have an Inverse? Let’s review the definition of the Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

  15. INVERSE The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

  16. Does the Matrix have an Inverse? Find the determinant! Therefore,NOinverse! Jeff Bivin -- LZHS

  17. Does the Matrix have an Inverse? Find the determinant! Therefore, an inverse exists! Jeff Bivin -- LZHS

  18. Does the Matrix have an Inverse? + + Find the determinant! - - - 1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2 45 + 84 + 96 - 105 - 48 - 72 0 Therefore,NOinverse! Jeff Bivin -- LZHS

  19. Does the Matrix have an Inverse? + + Find the determinant! - - - 1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3 8 + 9 + 16 - 24 - 4 - 12 -7 Therefore, an inverse exists! Jeff Bivin -- LZHS

  20. Solve the system using inverse matrices A U B 3x + 2y = 7 4x - 5y = 11 This is our Formula! Jeff Bivin -- LZHS

  21. Solve the system using inverse matrices A U B 2x - 4y = 9 3x - 2y = 1 This is our Formula! Jeff Bivin -- LZHS

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