SECTION 11.4 Matrix Algebra 805 Matrix (3) is in reduced row echelon form. Now reverse the earlier step of combining the two augmented matrices in (2), and write the single matrix (3) as two augmented matrices. − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 0 1 1 2 and 1 0 0 1 1 3 The conclusion from these matrices is that x z 1, 2, = = − and y w 1, 3. = − = Substituting these values into matrix (1) results in = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −A 1 1 2 3 1 Notice in the augmented matrix (3) that the 2 by 2 matrix to the right of the vertical bar is the inverse of A. Also notice that the identity matrix I2 appears to the left of the vertical bar. In general, using row operations to transform a nonsingular square matrix A, augmented by an identity matrix of the same dimensions, into reduced row echelon form results in the inverse matrix A .1− Steps for Finding the Inverse of an n by n Nonsingular Matrix A Step 1 Form the augmented matrix [ ] A I . n Step 2 Transform the matrix [ ] A In into reduced row echelon form. Step 3 The reduced row echelon form of [ ] A In contains the identity matrix In on the left of the vertical bar; the n by n matrix on the right of the vertical bar is the inverse of A. In Words If A is nonsingular, begin with the matrix [ ] A I , n and after transforming it into reduced row echelon form, you end up with the matrix ⎡ ⎣ ⎤ ⎦ − I A . n 1 Finding the Inverse of a Matrix The matrix A 1 1 0 1 3 4 0 4 3 = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ is nonsingular. Find its inverse. EXAMPLE 13 Algebraic Solution First, form the matrix [ ] = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ A I 1 1 0 1 3 4 0 4 3 1 0 0 0 1 0 0 0 1 3 Next, use row operations to transform [ ] A I3 into reduced row echelon form. 1 1 0 1 3 4 0 4 3 1 0 0 0 1 0 0 0 1 1 1 0 0 4 4 0 4 3 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 4 3 1 0 0 1 4 1 4 0 0 0 1 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ → ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ → ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ R r r 2 1 2 ↑ = + R r 1 4 2 2 ↑ = → − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 1 0 1 0 1 1 0 0 1 − − − ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 3 4 1 4 0 1 4 1 4 0 1 1 1 → − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 1 0 1 0 1 1 0 0 1 − − ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 3 4 1 4 0 1 4 1 4 0 1 1 1 Graphing Solution Enter the matrix A into a graphing utility. Figure 16 shows A 1− using a TI-84 Plus CE. (continued) Figure 16 Inverse matrix R r r R r r 1 4 1 2 1 3 2 3 ↑ =− + =− + R r1 3 3 ↑ =−
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