What is A(\text{adj } A) equal to?

Direction: Consider the following for the two (02) items that follow :<br>Let A=\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}

  1. A. \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}
  2. B. \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}
  3. C. \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/2 \end{bmatrix}
  4. D. \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ✓

Correct Answer: D. \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Explanation

We know the property A(\text{adj } A) = |A|I. Expanding the determinant of A: |A| = 3(-3(1) - 4(-1)) - (-3)(2(1) - 0) + 4(2(-1) - 0) = 3(1) + 6 - 8 = 1. Thus, A(\text{adj } A) = 1 \cdot I = I.

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