Let $$ \mathbf {X’X=}\begin {pmatrix} 4 & 2 & 1 \ 2 & 2 & 0 \ 1 & 0 & 1 \end {pmatrix}.$$Which one of the following matrices is $(\mathbf {X}’\mathbf {X})^{-1}$? (Hint: You do not need to know how to calculate an inverse of a matrix, you only need to check which of the following satisfies the definition of an inverse.)

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BlackTom AI · Accepted Answer

Question restatement: We are given X'X = [[4,2,1],[2,2,0],[1,0,1]] and asked which of the following matrices is its inverse, (X'X)^{-1}. The provided answer option is: a. [[1.0, -1.0, -1.0], [-1.0, 1.5, 1.0], [-1.0, 1.0, 2.0]].

Step-by-step verification approach: Since we are told not to compute an inverse directly, we can check whether the candidate B satisfies (X'X)B = I by straightforward matrix multiplication. If the product equals the identity matrix I, then B is indeed the inverse.

Compute A = X'X = [[4,2,1],[2,2,0],[1,0,1]] and B = [[1,-1,-1],[-1,1.5,1],[-1,1,2]]. We multiply A by B column by column.

- First column of B is [1, -1, -1]. A * first column =
  [4,2,1]·[1,-1,-1] = 4(1) + 2(-1) + 1(-1) = 4 - 2 - 1 = 1 (top entry),
  [2,2,0]·[1,-1,-1] = 2(1) + 2(-1) + 0(-1) = 2 - 2 + 0 = 0 (middle),
  [1,0,1]·[1,-1,-1] = 1(1) + 0(-1) + 1(-1) = 1 + 0 - 1 = 0 (bottom).
  So the first column of A*B is [1,0,0]^T.

- Second column of B is [-1, 1.5, 1]. A * second column =
  [4,2,1]·[-1,1.5,1] = 4(-1) + 2(1.5) + 1(1) = -4 + 3 + 1 = 0,
  [2,2,0]·[-1,1.5,1] = 2(-1) + 2(1.5) + 0(1) = -2 + 3 + 0 = 1,
  [1,0,1]·[-1,1.5,1] = 1(-1) + 0(1.5) + 1(1) = -1 + 0 + 1 = 0.
  So the second column of A*B is [0,1,0]^T.

- Third column of B is [-1, 1, 2]. A * third column =
  [4,2,1]·[-1,1,2] = 4(-1) + 2(1) + 1(2) = -4 + 2 + 2 = 0,
  [2,2,0]·[-1,1,2] = 2(-1) + 2(1) + 0(2) = -2 + 2 + 0 = 0,
  [1,0,1]·[-1,1,2] = 1(-1) + 0(1) + 1(2) = -1 + 0 + 2 = 1.
  So the third column of A*B is [0,0,1]^T.

Collectively, A*B = I, since we obtained columns that form [ [1,0,0], [0,1,0], [0,0,1] ]. Therefore, the candidate matrix B is indeed (X'X)^{-1}.

Additional consistency notes: The inverse is symmetric for a symmetric positive definite A like X'X, and the candidate B is symmetric as well, matching expectations for (X'X)^{-1}.

Conclusion (inference from calculation): The provided matrix in option a satisfies the inverse condition, since multiplying X'X by this matrix yields the identity matrix. Any other options (if they existed) would fail at least one entry in the product A*B, violating the definition of an inverse.

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