Question 23 of 58intermediate🔧 ApplyShort Answer2 marks

Find the inverse of each of the matrices (if it exists):

[10033052−1]\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}
Correct Answer

[100−1130323−1]\begin{bmatrix} 1 & 0 & 0 \\ -1 & \frac{1}{3} & 0 \\ 3 & \frac{2}{3} & -1 \end{bmatrix}

Exercise: EXERCISE 4.4 | Q: 8 | (Page 18)
For More Understanding

Explanation

To find the inverse, we first calculate the determinant to ensure the matrix is non-singular. Then, we find the cofactors of each element. The adjoint of the matrix is the transpose of the cofactor matrix. Finally, the inverse is calculated using the relation A⁻¹ = (adj A) / |A|, a method derived from the property A(adj A) = |A|I mentioned in the context.

Solution Steps

  1. Step 1: Calculate the determinant |A|. Expanding along Row 1: |A| = 1[(3)(-1) - (0)(2)] - 0 + 0 = 1(-3) = -3. Since |A| ≠\neq 0, the inverse exists.

  2. Step 2: Calculate the cofactors. A₁₁=-3, A₁₂=3, A₁₃=-9; A₂₁=0, A₂₂=-1, A₂₃=-2; A₃₁=0, A₃₂=0, A₃₃=3.

  3. Step 3: Form the adjoint matrix. Transpose the cofactor matrix: adj A = [−3003−10−9−23]\begin{bmatrix} -3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3 \end{bmatrix}.

  4. Step 4: Apply the inverse formula A⁻¹ = (1/|A|)(adj A). A⁻¹ = (1/-3) [−3003−10−9−23]=[100−1130323−1]\begin{bmatrix} -3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ -1 & \frac{1}{3} & 0 \\ 3 & \frac{2}{3} & -1 \end{bmatrix}.

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