M1410803 - (Section 8.3: The Inverse of a Square Matrix)...

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(Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1 . a 1 , or 1 a , is the multiplicative inverse of a , because its product with a is 1, the multiplicative identity. Example 3 1 3 = 1 , so 3 and 1 3 are multiplicative inverses of each other. PART B: THE INVERSE OF A SQUARE MATRIX If A is a square n × n matrix, sometimes there exists a matrix A 1 (“ A inverse”) such that AA 1 = I n and A 1 A = I n . An invertible matrix and its inverse commute with respect to matrix multiplication. Then, A is invertible (or nonsingular ), and A 1 is unique. In this course , an invertible matrix is assumed to be square. Technical Note: A nonsquare matrix may have a left inverse matrix or a right inverse matrix that “works” on one side of the product and produces an identity matrix. They cannot be the same matrix, however.
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(Section 8.3: The Inverse of a Square Matrix) 8.48 PART C: FINDING A - 1 We will discuss a shortcut for 2 × 2 matrices in Part F . Assume that A is a given n × n (square) matrix. A is invertible Its RRE Form is the identity matrix I n (or simply I ). It turns out that a sequence of EROs that takes you from an invertible matrix A down to I will also take you from I down to A 1 . (A good Linear Algebra book will have a proof for this.) We can use this fact to efficiently find A 1 . We construct A I . We say that A is in the “left square” of this matrix, and I is in the “right square.” We apply EROs to A I until we obtain the RRE Form I A 1 . That is, as soon as you obtain I in the left square, you grab the matrix in the right square as your A 1 . If you
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M1410803 - (Section 8.3: The Inverse of a Square Matrix)...

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