Question

jessica is finding the inverse of a 3x3 matrix using row reduction. she has started the problem. in which step has she made a mistake? a step 1 b step 2 c step 3 d step 4

Explanation:

Step1: Check Step 1

For \( R_2 - 2R_1 = R_2 \), original \( R_2 = [2,7,4,0,1,0] \), \( 2R_1 = [2,6,10,2,0,0] \). Subtracting: \( 2 - 2 = 0 \), \( 7 - 6 = 1 \), \( 4 - 10 = -6 \), \( 0 - 2 = -2 \), \( 1 - 0 = 1 \), \( 0 - 0 = 0 \). Matches the matrix in Step 1. So Step 1 is correct.

Step2: Check Step 2

For \( R_3 - 3R_1 = R_3 \), original \( R_3 = [3,5,8,0,0,1] \), \( 3R_1 = [3,9,15,3,0,0] \). Subtracting: \( 3 - 3 = 0 \), \( 5 - 9 = -4 \), \( 8 - 15 = -7 \), \( 0 - 3 = -3 \), \( 0 - 0 = 0 \), \( 1 - 0 = 1 \). But the matrix in Step 2 has \( R_3 = [0,-4,-7,0,0,1] \) (wait, the third element of the augmented part: original \( R_3 \) augmented is 0, after \( R_3 - 3R_1 \), it should be \( 0 - 3 = -3 \), but the matrix shows 0. Wait, no, let's re - check. Wait, the original augmented matrix for Step 2 before row operation: after Step 1, \( R_3 \) is still \( [3,5,8,0,0,1] \)? No, wait, after Step 1, the matrix is \( \begin{bmatrix}1&3&5&1&0&0\\0&1&-6&-2&1&0\\3&5&8&0&0&1\end{bmatrix} \). So \( R_3 - 3R_1 \): \( R_1 = [1,3,5,1,0,0] \), \( 3R_1 = [3,9,15,3,0,0] \). \( R_3 = [3,5,8,0,0,1] \), so \( 3 - 3 = 0 \), \( 5 - 9 = -4 \), \( 8 - 15 = -7 \), \( 0 - 3 = -3 \), \( 0 - 0 = 0 \), \( 1 - 0 = 1 \). But the matrix in Step 2 shows \( R_3 = [0,-4,-7,0,0,1] \) (the fourth element is 0, but should be - 3). Wait, maybe I messed up. Wait, let's check Step 3.

Step3: Check Step 3

The row operation is \( R_1 - 3R_2 = R_1 \). \( R_1 = [1,3,5,1,0,0] \), \( R_2 = [0,1,-6,-2,1,0] \), \( 3R_2 = [0,3,-18,-6,3,0] \). Subtracting: \( 1 - 0 = 1 \), \( 3 - 3 = 0 \), \( 5-(-18)=5 + 18 = 23 \), \( 1-(-6)=1 + 6 = 7 \), \( 0 - 3 = -3 \), \( 0 - 0 = 0 \). The matrix in Step 3 for \( R_1 \) is \( [1,0,23,7,-3,0] \), which is correct. Now check \( R_3 \) in Step 3: after Step 2, \( R_3 = [0,-4,-7,0,0,1] \), but after \( R_1 - 3R_2 \), \( R_3 \) should remain the same as Step 2? Wait, no, the row operation is only on \( R_1 \). Wait, the matrix in Step 3 shows \( R_3 = [0,-4,-7,-3,0,1] \). Wait, that's the mistake! In Step 3, the row operation is \( R_1 - 3R_2 = R_1 \), so only \( R_1 \) should change. But in the matrix for Step 3, \( R_3 \)'s fourth element (the augmented part) is - 3, which is wrong. Because \( R_3 \) was \( [0,-4,-7,0,0,1] \) in Step 2, and no operation was done on \( R_3 \) in Step 3. So Step 3 has a mistake. Wait, but let's confirm again. Wait, maybe I made a mistake in Step 2. Wait, Step 2: \( R_3 - 3R_1 = R_3 \). Original \( R_3 \) before Step 2 (after Step 1) is \( [3,5,8,0,0,1] \), \( 3R_1 = [3,9,15,3,0,0] \), so \( R_3 - 3R_1 = [0,-4,-7,-3,0,1] \). Oh! Wait, I see. In Step 2, the matrix after \( R_3 - 3R_1 = R_3 \) should be \( \begin{bmatrix}1&3&5&1&0&0\\0&1&-6&-2&1&0\\0&-4&-7&-3&0&1\end{bmatrix} \), but the matrix shown in Step 2 has \( R_3 = [0,-4,-7,0,0,1] \) (the fourth element is 0, but should be - 3). Wait, no, the user's Step 2 matrix is \( \begin{bmatrix}1&3&5&1&0&0\\0&1&-6&-2&1&0\\0&-4&-7&0&0&1\end{bmatrix} \). So Step 2 has \( R_3 \) with fourth element 0, but it should be - 3 (because \( 0 - 3=-3 \)). Wait, now I'm confused. Wait, let's start over.

Original augmented matrix: \( \begin{bmatrix}1&3&5&1&0&0\\2&7&4&0&1&0\\3&5&8&0&0&1\end{bmatrix} \)

Step 1: \( R_2 - 2R_1 = R_2 \)

\( R_2 \) becomes: \( [2 - 2(1),7 - 2(3),4 - 2(5),0 - 2(1),1 - 2(0),0 - 2(0)] = [0,1,-6,-2,1,0] \). Correct, as in Step 1's matrix.

Step 2: \( R_3 - 3R_1 = R_3 \)

\( R_3 \) becomes: \( [3 - 3(1),5 - 3(3),8 - 3(5),0 - 3(1),0 - 3(0),1 - 3(0)] = [0,-4,-7,-3,0,1] \). But the matrix in Step 2 shows \( R_3 = [0,-4,-7,0,0,1] \) (the fourth element is 0, should be - 3). Wait, no, the matrix in Step 2 as per the user is \( \begin{bmatrix}1&3&5&1&0&0\\0&1&-6&-2&1&0\\0&-4&-7&0&0&1\end{bmatrix} \). So Step 2 has an error? But then Step 3: \( R_1 - 3R_2 = R_1 \)

\( R_1 \) becomes: \( [1 - 3(0),3 - 3(1),5 - 3(-6),1 - 3(-2),0 - 3(1),0 - 3(0)] = [1,0,23,7,-3,0] \). Correct. Now, \( R_3 \) in Step 3's matrix is \( [0,-4,-7,-3,0,1] \), which is the correct result of Step 2 (since Step 2's matrix was wrong, but Step 3 uses the correct \( R_3 \) from Step 2's operation? No, the user's Step 2 matrix has \( R_3 \) with fourth element 0, but Step 3's \( R_3 \) has fourth element - 3. So the mistake is in Step 3? Wait, no. Wait, the row operation in Step 3 is only on \( R_1 \), so \( R_2 \) and \( R_3 \) should remain as in Step 2. But in Step 2, \( R_3 \) is \( [0,-4,-7,0,0,1] \), but in Step 3, \( R_3 \) is \( [0,-4,-7,-3,0,1] \). So Step 3 incorrectly changed \( R_3 \). Wait, this is confusing. Let's check the answer options. The options are Step 1, Step 2, Step 3, Step 4.

Wait, let's check Step 4: \( R_3 + 4R_2 = R_3 \). \( R_3 = [0,-4,-7,-3,0,1] \), \( 4R_2 = [0,4,-24,-8,4,0] \). Adding: \( 0 + 0 = 0 \), \( -4 + 4 = 0 \), \( -7-24=-31 \), \( -3-8=-11 \), \( 0 + 4 = 4 \), \( 1+0 = 1 \). Which matches the Step 4 matrix. So Step 4 is correct.

Step 1: correct. Step 2: Let's re - calculate \( R_3 - 3R_1 \). \( R_1 = [1,3,5,1,0,0] \), \( 3R_1 = [3,9,15,3,0,0] \), \( R_3 = [3,5,8,0,0,1] \). So \( R_3 - 3R_1 = [0,-4,-7,-3,0,1] \). But the matrix in Step 2 shows \( R_3 = [0,-4,-7,0,0,1] \) (the fourth element is 0, should be - 3). Wait, no, the matrix in Step 2 as per the user is \( \begin{bmatrix}1&3&5&1&0&0\\0&1&-6&-2&1&0\\0&-4&-7&0&0&1\end{bmatrix} \). So Step 2 has a mistake? But then Step 3: \( R_1 - 3R_2 = R_1 \). \( R_2 = [0,1,-6,-2,1,0] \), \( 3R_2 = [0,3,-18,-6,3,0] \), \( R_1 = [1,3,5,1,0,0] \). \( R_1 - 3R_2 = [1,0,23,7,-3,0] \), which is correct. Now, \( R_3 \) in Step 3's matrix is \( [0,-4,-7,-3,0,1] \), which is the correct result of \( R_3 - 3R_1 \) (even though Step 2's matrix was wrong). So Step 3 used the correct \( R_3 \) from the operation, but the Step 2 matrix was wrong. Wait, no, the problem is that in Step 3, the row operation is only on \( R_1 \), so \( R_3 \) should be the same as in Step 2. But in Step 2, \( R_3 \) is \( [0,-4,-7,0,0,1] \), and in Step 3, \( R_3 \) is \( [0,-4,-7,-3,0,1] \). So Step 3 incorrectly modified \( R_3 \). But according to the row operation \( R_1 - 3R_2 = R_1 \), only \( R_1 \) should change. So Step 3 has a mistake because \( R_3 \) was changed. Wait, but maybe the error is in Step 3's \( R_3 \). Alternatively, maybe the error is in Step 3's \( R_1 \)? No, \( R_1 \) calculation is correct. Wait, let's check the answer. The correct answer is Step 3 (Option C). Because in Step 3, the row operation is on \( R_1 \), so \( R_3 \) should not change from Step 2's \( R_3 \). But in Step 3's matrix, \( R_3 \) has a different fourth element (from 0 to - 3), which means \( R_3 \) was incorrectly modified.

Answer:

C. Step 3