Question

If matrix A = [a_ij]_2×2, where a_ij `{:(= 1  "if i" ≠ "j"),(= 0  "if i" = "j"):}` then A² is equal to ______.

Options

• I

• A

• 0

• None of these

Answer 1

If matrix A = [a_ij]_2×2, where a_ij `{:(= 1  "if i" ≠ "j"),(= 0  "if i" = "j"):}` then A² is equal to I.

Explanation:

Given that A = [a_ij]_2×2 

Let A = `[("a"_11, "a"_12),("a"_21, "a"_22)]_(2 x 2)`

a₁₁ = 0    ....[∵ i = j]

a₁₂ = 1    ....[∵ i ≠ j]

a₂₁ = 1    ....[∵ i ≠ j]

a₂₂ = 0    ....[∵ i = j]

∴ A = `[(0, 1),(1, 0)]`

Now, A² = A . A

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

=`[(0 + 1, 0 + 0),(0 +0, 1 + 0)]`

= I