Question

Find the inverse of `|(1, −1, 1),(2, 1, −3), (1, 1, 1)|` by adjoint method.

Answer 1

Given:

A = `|(1, −1, 1),(2, 1, −3), (1, 1, 1)|`

To find: A⁻¹ by the adjoint method.

Step 1: Find ∣A∣

`|A| = |(1, −1, 1),(2, 1, −3), (1, 1, 1)|`

Expanding along first row,

= 1 `|(1, −3), (1, 1)| − (−1) |(2, −3),(1, 1)| + 1 |(2, 1),(1, 1)|`

= 1(1 × 1 − (−3) × 1) + 1(2 × 1 − (−3) × 1) + 1(2 × 1 − 1 × 1)

= 1(4) + 1(5) + 1(1) = 10

∴ ∣A∣ = 10 ≠ 0

So,  A ⁻¹ exists.

Step 2: Find Cofactors Cij

Cofactor matrix C is:

C =  `|(C_11, C_12, C_13),(C_21, C_22, C_23), (C_31, C_32, C_33)| = |(4, −5, 1),(2, 0, −2), (2, 5, 3)|`

Step 3: Find adj(A)

adj(A) = C^T = `|(4, 2, 2),(−5, 0, 5), (1, −2, 3)|`

Step 4: Find A⁻¹

`A^(−1) = 1/|A|  "adj" (A) = 1/10 |(4, 2, 2),(−5, 0, 5), (1, −2, 3)|`

`A^(−1) = 1/10 |(4, 2, 2),(−5, 0, 5), (1, −2, 3)|`