If
[−4 3
−3 1]X+[2 −3
1 −5]=[−1 3
−2 4]X,
which of the following matrices is X?

Question

If
[−4	3
−3	1]X+[2	−3
1	−5]=[−1	3
−2	4]X,
which of the following matrices is X?

BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: Solve for X in the matrix equation [-4  3; -3  1] X + [2 -3; 1 -5] = [-1  3; -2  4] X.

Option under consideration (the only provided choice): X = [ [2/3, -1], [1/9, -4/3] ].

Step 1: Bring all X terms to one side. Subtract the right-hand side from the left-hand side to group like terms:
 AX + D = EX  =>  AX - EX = -D, where A = [-4 3; -3 1], D = [2 -3; 1 -5], E = [-1 3; -2 4].
 So (A - E) X = -D.

Step 2: Compute the matrix (A - E).
 A - E = [(-4) - (-1), 3 - 3; (-3) - (-2), 1 - 4] = [-3, 0; -1, -3].

Step 3: Check invertibility and find (A - E)^{-1}.
 Det(A - E) = (-3)(-3) - (0)(-1) = 9, which is nonzero, so the inverse exists.
 For a 2x2 matrix [a b; c d], its inverse is (1/det) [d, -b; -c, a]. Here a = -3, b = 0, c = -1, d = -3, so
 (A - E)^{-1} = (1/9) [ -3, 0; 1, -3 ].

Step 4: Solve for X using X = (A - E)^{-1} (-D).
 Compute -D = [ -2, 3; -1, 5 ].
 Multiply (A - E)^{-1} by (-D):
 (A - E)^{-1} (-D) = (1/9) [ -3, 0; 1, -3 ] [ -2, 3; -1, 5 ].
 First, multiply the two 2x2 matrices: 
 [ -3, 0; 1, -3 ] * [ -2, 3; -1, 5 ] = [ (-3)(-2) + 0(-1), (-3)(3) + 0(5); (1)(-2) + (-3)(-1), (1)(3) + (-3)(5) ]
 = [ 6, -9; -2 + 3, 3 - 15 ] = [ 6, -9; 1, -12 ].
 Now apply the factor (1/9): X = (1/9) [ 6, -9; 1, -12 ].
 This yields X = [ [6/9, -9/9], [1/9, -12/9] ] = [ [2/3, -1], [1/9, -4/3] ].

Step 5: Compare with the provided option.
 The computed X matches the given candidate exactly: X = [ [2/3, -1], [1/9, -4/3] ].

Note on other options: There are no additional answer choices provided in the prompt to evaluate. If other options existed, we would test each by substituting back into the original equation or by repeating the algebraic steps to verify whether AX + D equals EX for that X. In this scenario, the single given option is consistent with the algebraic solution, and thus is the correct X.

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

If [−4 3 −3 1]X+[2 −3 1 −5]=[−1 3 −2 4]X, which of the following matrices is X?

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