Question

If A = `[(1, 2),(3, 4)]`, prove that A · (adj A) = (adj A) · A = |A| · I

Answer 1

Given: A = `[(1, 2),(3, 4)]`

Step 1: Find |A| (Determinant)

|A| = (1)(4) – (2)(3)

= 4 – 6

= –2

Step 2: Find adj A

For a 2 × 2 matrix

A = `[(a, b),(c, d)]`

adj A = `[(d, -b),(-c, a)]`

So, adj A = `[(4, -2),(-3, 1)]`

Step 3: Find A · adj A

A · adj A = `[(1, 2),(3, 4)] [(4, -2),(-3, 1)]`

Multiply:

First row:

(1)(4) + (2)(–3)

= 4 – 6

= –2

(1)(–2) + (2)(1)

= –2 + 2

= 0

Second row:

(3)(4) + (4)(–3)

= 12 – 12

= 0

(3)(–2) + (4)(1)

= –6 + 4

= –2

So, A · adj A = `[(-2, 0),(0, -2)]`

= `-2[(1, 0),(0, 1)]`

= |A| I

Step 4: Find adj A · A

`[(4, -2),(-3, 1)][(1, 2),(3, 4)]`

First row:

(4)(1) + (–2)(3)

= 4 – 6

= –2

(4)(2) + (–2)(4)

= 8 – 8

= 0

Second row:

(–3)(1) + (1)(3)

= –3 + 3

= 0

(–3)(2) + (1)(4)

= –6 + 4

= –2

adj A · A = `[(-2, 0),(0, -2)]` = |A| I

Hence Proved

A · (adj A) = (adj A) · A = |A| · I for the given matrix.