If
[
−
1

1

5

5
]
𝑋
+
[
4

−
1

2

−
5
]
=
[
1

0

1

−
4
]
𝑋
,
which of the following matrices is
𝑋
?

Question

If
[
−
1
	
1

5
	
5
]
𝑋
+
[
4
	
−
1

2
	
−
5
]
=
[
1
	
0

1
	
−
4
]
𝑋
,
which of the following matrices is 
𝑋
?

BlackTom AI · Accepted Answer

Subtracting C X from both sides gives (A − C)X = −B where A = [[−1, 1],[5, 5]], B = [[4, −1],[2, −5]], C = [[1, 0],[1, −4]]. Key concepts: matrix addition/subtraction, left-multiplication by a matrix, and solving for X via the inverse of a non-singular coefficient matrix.

1) Compute A − C:
A − C = [[−1 − 1, 1 − 0],[5 − 1, 5 − (−4)]] = [[−2, 1],[4, 9]] = D.

2) Compute −B:
−B = [[−4, 1],[−2, 5]].

3) Solve D X = −B by X = D^{-1}(−B). Compute det(D):
det(D) = (−2)(9) − (1)(4) = −18 − 4 = −22 (nonzero, so inverse exists).

4) Compute D^{-1}:
D^{-1} = (1/det) [[d, −b],[−c, a]] = (1/−22) [[9, −1],[−4, −2]].

5) Multiply D^{-1} by −B. First compute M = [[9, −1],[−4, −2]] · [[−4, 1],[−2, 5]]:
- M[1,1] = 9(−4) + (−1)(−2) = −36 + 2 = −34
- M[1,2] = 9(1) + (−1)(5) = 9 − 5 = 4
- M[2,1] = (−4)(−4) + (−2)(−2) = 16 + 4 = 20
- M[2,2] = (−4)(1) + (−2)(5) = −4 − 10 = −14
So M = [[−34, 4],[20, −14]].

6) Multiply by 1/−22:
X = (1/−22)M = [[(−34)/(−22), 4/(−22)],[20/(−22), (−14)/(−22)]]
Simplify fractions: = [[34/22, −4/22], [−20/22, 14/22]] = [[17/11, −2/11], [−10/11, 7/11]].

Why this is correct: direct algebraic manipulation and explicit inversion yield X = [[17/11, −2/11],[−10/11, 7/11]], which satisfies (A − C)X = −B and thus the original equation A X + B = C X.

Why other options are incorrect:
- Option 1 [[−4/11, −2/11],[3/11, −4/11]] does not satisfy the derived linear system (would produce different entries when substituted).
- Option 2 integer matrix [[−3, −2],[−8, −7]] is far from the fractional solution and fails the equation.
- Option 4 [[−1/7, −1/7],[1/7, −5/14]] has different entries and does not satisfy (A − C)X = −B.

Final confirmation: the correct matrix X is [[17/11, −2/11],[−10/11, 7/11]], matching the provided answer.

If [ − 1 1 5 5 ] 𝑋 + [ 4 − 1 2 − 5 ] = [ 1 0 1 − 4 ] 𝑋 , which of the following matrices is 𝑋 ?Single choice

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