If
𝑀
=
[
1

−
7

9

0

1

−
4

0

0

1
]
,
then which of the following is
𝑀
−
1
?

Question

If
𝑀
=
[
1
	
−
7
	
9

0
	
1
	
−
4

0
	
0
	
1
]
,
then which of the following is 
𝑀
−
1
?

BlackTom AI · Accepted Answer

We start by identifying the given matrix M. It is a 3x3 upper-triangular matrix with ones on the diagonal:
M = [ [1, -7, 9], [0, 1, -4], [0, 0, 1] ]
We seek M^{-1}, which for a unit upper-triangular matrix has the form:
M^{-1} = [ [1, a, b], [0, 1, c], [0, 0, 1] ]
So we compute the product M · M^{-1} and set it equal to the identity to solve for a, b, c.
Compute the product:
First row: [1, -7, 9] · column 1 = 1, as required; column 2 gives a − 7, which must be 0, so a = 7; column 3 gives b − 7c + 9, which must be 0, so b = 7c − 9.
Second row: [0, 1, −4] · column 3 gives c − 4, which must be 0, so c = 4.
Then b = 7·4 − 9 = 28 − 9 = 19.
Thus M^{-1} = [ [1, 7, 19], [0, 1, 4], [0, 0, 1] ].
Now compare this result to each option:
Option 1: [ [1, 7, 18], [0, 1, 4], [0, 0, 1] ] — incorrect because the (1,3) entry is 18, but it should be 19.
Option 2: [ [1, 7, 26], [0, 1, 5], [0, 0, 1] ] — incorrect because a = 7 is fine, but c = 5 is wrong (should be 4), and the (1,3) entry is 26, not 19.
Option 3: [ [1, 6, 15], [0, 1, 4], [0, 0, 1] ] — incorrect because a should be 7, not 6; also (1,3) = 15 is wrong.
Option 4: [ [1, 7, 19], [0, 1, 4], [0, 0, 1] ] — correct, as derived.
In summary, the key steps are identifying the unit upper-triangular structure, setting up M^{-1} with unknowns a, b, c, solving the equations from M · M^{-1} = I, obtaining a = 7, c = 4, b = 19, and verifying against each option.

If 𝑀 = [ 1 − 7 9 0 1 − 4 0 0 1 ] , then which of the following is 𝑀 − 1 ?

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