Question 53 of 58intermediate🔧 ApplyLong Answer4 marks

Let A = [121231115]\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}. Verify that (i) [adj A]^{-1} = adj (A−1A^{-1}) (ii) (A−1A^{-1})^{-1} = A

(i)
Answer

[adj A]^{-1} = adj (A−1A^{-1}) is verified as both equal to [−1/5−2/5−1/5−2/5−3/5−1/5−1/5−1/5−1]\begin{bmatrix} -1/5 & -2/5 & -1/5 \\ -2/5 & -3/5 & -1/5 \\ -1/5 & -1/5 & -1 \end{bmatrix}

(ii)
Answer

(A−1A^{-1})^{-1} = A is verified as (A−1A^{-1})^{-1} = [121231115]=A\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix} = A

Explanation

This question verifies two important properties of invertible matrices. For part (i), we use the property that [adj A]^{-1} = (1/|adj A|) × adj(adj A) and adj(A−1A^{-1}) = A/|A|. For part (ii), we use the fundamental property that the inverse of the inverse gives back the original matrix.

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