02-12-03 - Lecture 6 Andrei Antonenko February 12, 2003 1...

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Lecture 6 Andrei Antonenko February 12, 2003 1 Multiplicative inverse In this lecture we will continue with the properties of matrix operations. (M4) Existence of the multiplicative inverse. Matrix B is called the inverse for the square matrix A if BA = AB = I . The existence of the inverse is a very difficult question, which we will solve later. Now we’ll simply give some examples, and then a formula for an inverse of 2 × 2-matrices. Example 1.1. The matrix ˆ 1 0 0 0 ! doesn’t have an inverse, since if ˆ a b c d ! is an inverse, than ˆ 1 0 0 0 a b c d ! = ˆ 1 0 0 1 ! , from what ˆ a b 0 0 ! = ˆ 1 0 0 1 ! which is never the case. Example 1.2. The inverse for the matrix ˆ 1 2 3 5 ! is ˆ - 5 2 3 - 1 ! since ˆ 1 2 3 5 - 5 2 3 - 1 ! = ˆ 1 0 0 1 ! Now we’re ready to give a formula for an inverse of 2 × 2-matrix. Proposition 1.3. The inverse of 2 × 2 -matrix ˆ a b c d ! exists if and only if ad - bc 6 = 0 and ˆ a b c d ! - 1 = 1 ad - bc ˆ d - b - c a ! Proof. If ad - bc 6 = 0 then we can simply check that this matrix is the inverse: ˆ a b c d ! × 1 ad - bc ˆ d - b - c a ! = 1 ad - bc ˆ ad - bc - ab + ab cd - cd - cb + da ! = ˆ 1 0 0 1 ! 1
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We will not give a proof that if a matrix has an inverse, then ad - bc 6 = 0. This fact can be generalized to the case of larger matrices, and we’ll prove it later in more general form. The proposition above is useful when you want to get an inverse of a 2 × 2-matrix. Later we’ll provide a method of finding the inverses of larger matrices, but for 2 × 2-matrices this is the easiest one. 2 Transpose of a matrix Definition 2.1. Matrix B is called transpose of a matrix A (notation: B = A > ) if b ij = a ji . In other words, we should take rows of a matrix A and write them as columns of matrix B . Then B = A > . In general form, if A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn then A > = a 11 a 21 ··· a m 1 a 12 a 22 ··· a m 2 . . . . . . . . . . . . a 1 n a 2 n ··· a mn Let’s notice, that if A is an m × n -matrix, then A > is an n × m -matrix. Example 2.2. Let A = ˆ 1 2 3 4 ! . Than A > = ˆ 1 3 2 4 ! . Example 2.3.
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02-12-03 - Lecture 6 Andrei Antonenko February 12, 2003 1...

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