804 CHAPTER 11 Systems of Equations and Inequalities Not every square matrix has an inverse. When a matrix A has an inverse A ,1− then A is said to be nonsingular. If a matrix A has no inverse, it is called singular. Multiplying a Matrix by Its Inverse Show that the inverse of = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − A A 3 1 2 1 is 1 1 2 3 1 EXAMPLE 12 Solution We need to show that = = − − AA A A I . 1 1 2 = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − AA I A A I 3 1 2 1 1 1 2 3 1 0 0 1 1 1 2 3 3 1 2 1 1 0 0 1 1 2 1 2 The following shows one way to find the inverse of = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A 3 1 2 1 Suppose that A 1− is given by = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −A x y z w 1 (1) where x, y, z, and w are four variables. Based on the definition of an inverse, if A has an inverse, then AA I x y z w x z y w x z y w 3 1 2 1 1 0 0 1 3 3 2 2 1 0 0 1 1 2 = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + + + + ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − Because corresponding entries must be equal, it follows that this matrix equation is equivalent to two systems of linear equations. + = + = ⎧ ⎨ ⎪⎪ ⎩⎪⎪ + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x z x z y w y w 3 1 2 0 3 0 2 1 The augmented matrix of each system is 3 1 2 1 1 0 3 1 2 1 0 1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (2) The usual procedure would be to transform each augmented matrix into reduced row echelon form. Notice, though, that the left sides of the augmented matrices are equal, so the same row operations (see Section 11.2) can be used to reduce both matrices. It is more efficient to combine the two augmented matrices (2) into a single matrix, as shown next. ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 3 1 2 1 1 0 0 1 Now, use row operations to transform the matrix into reduced row echelon form. ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ → − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ → − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 3 1 2 1 1 0 0 1 1 0 2 1 1 1 0 1 1 0 0 1 1 1 2 3 (3) ↑ R r r 1 1 2 1 =− + ↑ =− + R r r 2 2 1 2
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