Consider the matrix: $$ \mathbf {X=}\begin {pmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 1 & 0 \ 1 & 0 & 1 \end {pmatrix}.$$Which one of the following matrices is $\mathbf {X}’\mathbf {X}$?

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BlackTom AI · Accepted Answer

We start by understanding the task: compute X'X for the given 4-by-3 matrix X.

First, identify the columns of X, since X'X = X^T X uses inner products of columns. The columns are:
- col1 = [1, 1, 1, 1]^T
- col2 = [0, 1, 1, 0]^T
- col3 = [0, 0, 0, 1]^T

Next, compute the necessary dot products to fill the 3-by-3 matrix X^T X:
- col1 · col1 = 1^2 + 1^2 + 1^2 + 1^2 = 4
- col1 · col2 = 1*0 + 1*1 + 1*1 + 1*0 = 2
- col1 · col3 = 1*0 + 1*0 + 1*0 + 1*1 = 1
- col2 · col2 = 0^2 + 1^2 + 1^2 + 0^2 = 2
- col2 · col3 = 0*0 + 1*0 + 1*0 + 0*1 = 0
- col3 · col3 = 0^2 + 0^2 + 0^2 + 1^2 = 1

Assemble these into the symmetric X^T X matrix using the computed dot products:
X^T X = [[4, 2, 1],
         [2, 2, 0],
         [1, 0, 1]]

Now compare with the provided options. The matrix in option d is exactly
[[4, 2, 1],
 [2, 2, 0],
 [1, 0, 1]]
which matches the computed X^T X.

If we were to examine other hypothetical options (a, b, c), we would check whether their entries correspond to the same dot-product calculations. Any deviation in a single entry would indicate a mismatch with the actual column-wise inner products, signaling an error in that alternative.

In summary, by taking the inner products of the columns of X and assembling them into the 3-by-3 matrix, the resulting X^T X is precisely the matrix shown in option d.

ETC2410 - ETC5241 - BEX2410 - S2 2025 Tutorial 4: Part A Quiz

Consider the matrix: \[ \mathbf {X=}\begin {pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end {pmatrix}.\]Which one of the following matrices is \(\mathbf {X}’\mathbf {X}\)?

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ETC2410 - ETC5241 - BEX2410 - S2 2025 Tutorial 4: Part A Quiz

Consider the matrix: \[ \mathbf {X=}\begin {pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end {pmatrix}.\]Which one of the following matrices is \(\mathbf {X}’\mathbf {X}\)?

View Explanation

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If A=[54−34−1−300−5], then which of the following is A2?

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Vector a = 10 12 And vector b = 2 3 Whats the value of ab ? ( is matrix multiplication); Please enter a number with no decimals.