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Identity and Inverse Matrices

Identity and Inverse Matrices. Section 7.2b!!!. First, New Definitions:. The n x n matrix I with 1’s on the main diagonal (upper left to lower right) and 0’s elsewhere is the identity matrix of order n x n :. n. First, New Definitions:.

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Identity and Inverse Matrices

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  1. Identity and Inverse Matrices Section 7.2b!!!

  2. First, New Definitions: The n x n matrix I with 1’s on the main diagonal (upper left to lower right) and 0’s elsewhere is the identity matrix of order n x n: n

  3. First, New Definitions: The n x n matrix I with 1’s on the main diagonal (upper left to lower right) and 0’s elsewhere is the identity matrix of order n x n: n For example…

  4. First, New Definitions: If is any n x n matrix, then That is, is the multiplicative identity for the set of n x n matrices.

  5. Definition:Inverse of a Square Matrix Let be an n x n matrix. If there is a matrix B such that –1 then B is the inverse of A. We write B = A (read “A inverse”) Important note: Not every square matrix has an inverse. If a square matrix A has an inverse, then A is nonsingular. If A has no inverse, then A is singular.

  6. Practice Problems!!! Prove (both algebraically and using a calculator) that the given matrices are inverses.

  7. Practice Problems!!! Prove that the given matrix is singular. If B has an inverse, then AB = I 2 Take a look at these top two equations…

  8. Determinant of a Square Matrix

  9. Consider this: We can always use the calculator, but there is also an Algebraic method for solving… Inverse of a 2 x 2 Matrix If , then

  10. Consider this: Verify with your calculator!!!

  11. Definitions The number ad – bc is the determinant of the 2 x 2 matrix: …and is denoted:

  12. Definitions To define the determinant for higher order square matrices, we need a couple more definitions… The minor (short for “minor determinant”) M corresponding to a given element is the determinant of the part of the matrix obtained by deleting the row and column containing that element. ij The cofactor corresponding to that element is

  13. Definitions Determinant of a Square Matrix Let A = [a ] be a matrix of order n x n (n > 2). The determinant of A, denoted by det A or |A|, is the sum of the entries in any row or any column multiplied by their respective cofactors. For example, expanding the i-th row gives ij

  14. Apply the Definition: Use the definition to evaluate the determinant of B: Calculator verification?!?!

  15. One more theorem… Inverses of n x n Matrices An n x n matrix has an inverse if and only if det A = 0. Note: There are more complicated formulas for finding the inverses of matrices of order 3 x 3 or higher, but in this course, we will simply use a calculator in these cases…

  16. Guided Practice Determine algebraically whether the matrix has an inverse. If so, find its inverse matrix. Can we verify this answer using the definition of inverses???

  17. Guided Practice Determine algebraically whether the matrix has an inverse. If so, find its inverse matrix.

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