If
M=[1 0 0
0 1 0
6 3 1],
then which of the following is M−1?

Question

If
M=[1	0	0
0	1	0
6	3	1],
then which of the following is M−1?

BlackTom AI · Accepted Answer

Question restatement: Given M = [ [1, 0, 0], [0, 1, 0], [6, 3, 1] ], determine M^{-1}.

First, observe the structure of M. It is a lower triangular matrix with 1s on its diagonal. Such matrices are unimodular and have inverses that are also lower triangular with 1s on the diagonal.

Step-by-step method to find M^{-1}:
- One way to verify a candidate inverse N is to check that M * N = I. Since M has a simple form, we can try to compute N directly by solving systems M x_j = e_j for each column j of the identity, where e_j is the j-th standard basis vector.
- Here, we test the provided candidate inverse N = [ [1, 0, 0], [0, 1, 0], [-6, -3, 1] ]. We will multiply M by N to see if the product is the identity.

Matrix multiplication check: compute M * N.
- Row 1 of M times columns of N: Row 1 is [1, 0, 0]. Multiplying by N yields [1, 0, 0], which matches the first row of I.
- Row 2 of M times columns of N: Row 2 is [0, 1, 0]. Multiplying by N yields [0, 1, 0], which matches the second row of I.
- Row 3 of M times columns of N: Row 3 is [6, 3, 1]. Compute the products with each column of N:
  • Column 1 of N is [1, 0, -6]^T, so 6*1 + 3*0 + 1*(-6) = 6 + 0 - 6 = 0.
  • Column 2 of N is [0, 1, -3]^T, so 6*0 + 3*1 + 1*(-3) = 0 + 3 - 3 = 0.
  • Column 3 of N is [0, 0, 1]^T, so 6*0 + 3*0 + 1*1 = 1.
  This yields the row [0, 0, 1], which matches the third row of I.

Since M * N equals the identity matrix, the candidate N is indeed M^{-1}.

Additional intuition: For a lower triangular matrix with ones on the diagonal, the inverse can be obtained by negating the appropriate combinations that reproduce the identity; here, the entries -6 and -3 in the (3,1) and (3,2) positions accomplish the necessary cancellations when multiplying by M.

Conclusion about the provided candidate: The matrix [[1, 0, 0], [0, 1, 0], [-6, -3, 1]] is the inverse of M because it satisfies M * M^{-1} = I. If there were multiple-choice options, this would be the correct choice; however, note that no explicit answer options were provided in the prompt to compare against.

MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 8

If M=[1 0 0 0 1 0 6 3 1], then which of the following is M−1?

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