If $$M = \left [\begin{matrix}1 & 0 & 0\-9 & 1 & 0\6 & 2 & 1\end{matrix}\right ],$$ then which of the following is $M^{-1}$?

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BlackTom AI · Accepted Answer

We are given M = [ [1, 0, 0], [-9, 1, 0], [6, 2, 1] ] and asked for M^{-1}.
Since M is a 3x3 unit lower triangular matrix (ones on the diagonal), its inverse will also be unit lower triangular, of the form:
N = [ [1, 0, 0], [a, 1, 0], [b, c, 1] ].
We want MN = I, so we solve for a, b, c.
Compute the product MN:
- Row 1 of MN is [1, 0, 0], which already matches the identity row [1, 0, 0].
- For row 2: Multiply row 2 of M by columns of N.
  Col 1 gives (-9)*1 + 1*a + 0*b = -9 + a. We need this to be 0, so a = 9.
  Col 2 gives (-9)*0 + 1*1 + 0*c = 1, which matches the identity value 1. Col 3 gives (-9)*0 + 1*0 + 0*1 = 0, matching the identity 0.
- For row 3:
  Col 1 gives 6*1 + 2*a + 1*b = 6 + 2a + b. We need this to be 0. With a = 9, this is 6 + 18 + b = 0, so b = -24.
  Col 2 gives 6*0 + 2*1 + 1*c = 2 + c. We need this to be 0, so c = -2.
  Col 3 gives 6*0 + 2*0 + 1*1 = 1, which matches the identity 1.
Thus the inverse is:
M^{-1} = [ [1, 0, 0], [9, 1, 0], [-24, -2, 1] ].
This exactly matches the provided option: [ [1, 0, 0], [9, 1, 0], [-24, -2, 1] ].
If there were other options, we would verify each by computing MN and comparing to I, but in this case the single listed candidate is the correct inverse and satisfies all row-by-column constraints required for the identity matrix.

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

If \[M = \left [\begin{matrix}1 & 0 & 0\\-9 & 1 & 0\\6 & 2 & 1\end{matrix}\right ],\] then which of the following is \(M^{-1}\)?

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