What is A^{-1} 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} 1 & -1 & 0 \\ -2 & 3 & -4 \\ -2 & 3 & -3 \end{bmatrix} ✓
  2. B. \begin{bmatrix} 1/2 & -1/2 & 0 \\ -1 & 3/2 & -2 \\ -1 & 3/2 & -3/2 \end{bmatrix}
  3. C. \begin{bmatrix} 2 & -2 & 0 \\ -4 & 6 & -8 \\ -4 & 6 & -6 \end{bmatrix}
  4. D. \begin{bmatrix} 1/5 & -1/5 & 0 \\ -2/5 & 3/5 & -4/5 \\ -2/5 & 3/5 & -3/5 \end{bmatrix}

Correct Answer: A. \begin{bmatrix} 1 & -1 & 0 \\ -2 & 3 & -4 \\ -2 & 3 & -3 \end{bmatrix}

Explanation

Since |A|=1, A^{-1} = \text{adj } A. Calculating the cofactors for the transposed matrix gives: C_{11} = (-3)(1) - 4(-1) = 1, C_{21} = -(-3(1) - 4(-1)) = -1, and C_{31} = (-3)(4) - (-3)(4) = 0. This forms the first row of \text{adj } A as \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}. Checking the options, only the first matrix matches this row.

Related questions on Matrices & Determinants

Practice more NDA Mathematics questions