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Lecture 22 - Mar 2

Lecture 22 - Mar 2 - Monday March 2 − Lecture 22 The...

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Unformatted text preview: Monday March 2 − Lecture 22 : The inverse of a square matrix. (Refers to section 3.5) Expectations: 1. Define invertible matrix. 2. Recognize the "uniqueness" of the inverse of a matrix. 3. Define a matrix to a negative power. 4. Prove that the product of the inverses (when these exist) is the inverse of a product of matrices (but in the reverse order). 5. Prove the transpose of the inverse of a matrix is the transpose of the inverse. 6. Prove [ cA ] − 1 = (1/ c ) A − 1 . 7. Recognize and use the formula for the inverse of a 2 by 2 matrix. 8. Define the determinant of a 2 × 2 matrix. 22.0 Notation − Given n an integer, we will use the notation , M n × n , (sometimes simply M n ) for the set of all square matrices of size n × n . That is, if the matrix A belongs to M n × n then it is an n × n matrix. 22.1 Definition − We say that a matrix A in M n , is invertible ( or non-singular ) if there exists a matrix B in M n × n such both conditions AB = I n and BA = I n . hold true. We call such a matrix B , an inverse of A . We note that here we are speaking only of square matrices. 22.1.1 Proposition − On " uniqueness of the inverse of a matrix " : Let A be an invertible matrix in M n × n . Then there exists no more than one inverse B of A in M n ....
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Lecture 22 - Mar 2 - Monday March 2 − Lecture 22 The...

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