If $$M = \left [\begin{matrix}1 & -2 & 3\0 & 1 & -1\0 & 0 & 1\end{matrix}\right ],$$ then which of the following is $M^{-1}$?

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Question restatement: If M = [ [1, -2, 3], [0, 1, -1], [0, 0, 1] ], then which of the following is M^{-1}?

Step-by-step analysis of the given information: The matrix M is upper triangular with all diagonal entries equal to 1, so it is a unit upper triangular matrix. For such matrices, the inverse is also upper triangular with 1s on the diagonal. We can determine the off-diagonal entries by solving M · M^{-1} = I.

Let M^{-1} have the form: [ [1, a, b], [0, 1, c], [0, 0, 1] ]. The product M · M^{-1} must equal the identity I = [ [1, 0, 0], [0, 1, 0], [0, 0, 1] ].

Compute the product block by block:
- First row, first column: 1·1 + (-2)·0 + 3·0 = 1, which matches I(1,1).
- First row, second column: 1·a + (-2)·1 + 3·0 = a - 2. To equal 0, we need a = 2.
- First row, third column: 1·b + (-2)·c + 3·1 = b - 2c + 3. To equal 0, we require b = 2c - 3.
- Second row, first column: 0·1 + 1·0 + (-1)·0 = 0, which is correct for I(2,1).
- Second row, second column: 0·a + 1·1 + (-1)·0 = 1, which matches I(2,2).
- Second row, third column: 0·b + 1·c + (-1)·1 = c - 1. To equal 0, we need c = 1.
- Third row, first and second columns trivially yield 0 because of zeros below the diagonal; third row, third column yields 1, as required.

From c = 1, we get b = 2c - 3 = 2·1 - 3 = -1. Therefore a = 2, b = -1, c = 1.

Thus M^{-1} = [ [1, 2, -1], [0, 1, 1], [0, 0, 1] ].

Review of the given option: The only provided option is
[ [1, 2, -1], [0, 1, 1], [0, 0, 1] ].
This matches the computed M^{-1} exactly, confirming it is correct.

Summary of why the other potential forms (if any were present) would be incorrect: any deviation in the upper-triangular entries would fail to satisfy M · M^{-1} = I in at least one position, generally breaking the 1s on the diagonal or the zero off-diagonal requirements. Since the matrix must maintain 1s on the diagonal for a unit upper triangular inverse, and the off-diagonal entries were constrained by the equations a = 2 and c = 1 with b = -1, only this configuration satisfies all conditions.

Note: The problem statement provided a single candidate option rather than a list of alternatives. The analysis above confirms that the given option is indeed M^{-1}.

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

If \[M = \left [\begin{matrix}1 & -2 & 3\\0 & 1 & -1\\0 & 0 & 1\end{matrix}\right ],\] then which of the following is \(M^{-1}\)?

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