Question

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M^TM is seven, is ______.

Options

• 540

• 541

• 542

• 543

Answer 1

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M^TM is seven, is 540.

Explanation:

Given: M is any 3 × 3 matrix with entries from set {0, 1, 2}.

Let M = `[("a", "b", "c"),("d", "e", "f"),("g", "h", "i")]`

M^T = `[("a", "d", "g"),("b", "e", "h"),("c", "f", "i")]`

M^TM = `[("a", "d", "g"),("b", "e", "h"),("c", "f", "i")][("a", "b", "c"),("d", "e", "f"),("g", "h", "i")]`

Sum of diagonal elements = 7

⇒ a² + b² + c² + d² + e² + f² + g² + h² + i² = 7

Case I: Seven (1's) and two (0's) can be used

∴ No. of ways = ⁹C₂ = 36 (Selection of 2 zero out of 7(1s) and 2(0's) or selection of 7(1's) from 7(1's) and 2(0's))

Similarly,

Case II: One (2's) and 3(1's) and (0's)

∴ ⁹C₁ × ⁸C₃ × ⁵C₅ = 504

∴ Total = 36 + 504 = 540