[iD] Diketahui A=[} 1&1&1 0&1&1 0&0&1 ] Tentukan pernyataan yang benar terkait A(A^-1B)^-1(B^-1A)^-1A [cN] Srd row [io] Baris ke -3 square [io] Kolom ke-3 square [EN] Ist row square [io] Baris ke-1 [EN] Ist column [io] Kolom ke -1 square

Question

Pertanyaan

[iD] Diketahui A=[} 1&1&1 0&1&1 0&0&1 ] Tentukan pernyataan yang benar terkait A(A^-1B)^-1(B^-1A)^-1A [cN] Srd row [io] Baris ke -3 square [io] Kolom ke-3 square [EN] Ist row square [io] Baris ke-1 [EN] Ist column [io] Kolom ke -1 square

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Answer 1

Let A and B be the given matrices:
$A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 3 & \frac{1}{4} \\ 0 & \frac{1}{9} & 7 \\ 3 & 1 & 1 \end{bmatrix}$

We want to find the expression $A(A^{-1}B)^{-1}(B^{-1}A)^{-1}A$. Let's simplify this expression using properties of matrix inverses.

Recall that $(XY)^{-1} = Y^{-1}X^{-1}$. Therefore:

$(A^{-1}B)^{-1} = B^{-1}A$ and $(B^{-1}A)^{-1} = A^{-1}B$.

Substituting these into the original expression, we get:

$A(A^{-1}B)^{-1}(B^{-1}A)^{-1}A = A(B^{-1}A)(A^{-1}B)A$

Now, we can use the associative property of matrix multiplication:

$A(B^{-1}A)(A^{-1}B)A = A(B^{-1}(AA^{-1})B)A = A(B^{-1}IB)A = A(B^{-1}B)A = AIA = A$

Therefore, $A(A^{-1}B)^{-1}(B^{-1}A)^{-1}A = A$.

The matrix A is:
$A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$

The question asks which statement is true about $A(A^{-1}B)^{-1}(B^{-1}A)^{-1}A = A$. Since we've shown this simplifies to A, any statement that is true about matrix A is correct.

Let's examine the options:

* Third row: The third row of A is [0 0 1].
* Third column: The third column of A is [1 1 1].
* First row: The first row of A is [1 1 1].
* First column: The first column of A is [1 0 0].

Therefore, the statements "Third row" and "First row" and "Third column" and "First column" are all potentially correct depending on what the question is asking for specifically. The question is ambiguous. It needs to specify what property of the resulting matrix A it is interested in.

Pertanyaan

[iD] Diketahui A=[} 1&1&1 0&1&1 0&0&1 ] Tentukan pernyataan yang benar terkait A(A^-1B)^-1(B^-1A)^-1A [cN] Srd row [io] Baris ke -3 square [io] Kolom ke-3 square [EN] Ist row square [io] Baris ke-1 [EN] Ist column [io] Kolom ke -1 square

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