# Solve the following : If A = [2-43-201],B=[1-12-210], then show that (AB)T = BTAT. - Mathematics and Statistics

Sum

Solve the following :

If A = [(2, -4),(3, -2),(0, 1)], "B" = [(1, -1, 2),(-2, 1, 0)], then show that (AB)T = BTAT.

#### Solution

AB = [(2, -4),(3, -2),(0, 1)] [(1, -1, 2),(-2, 1, 0)]

= [(2 + 8, -2 - 4, 4 + 0),(3 + 4, -3 - 2, 6 - 0),(0 - 2, 0 + 1, 0 + 0)]

= [(10, -6, 4),(7, -5, 6),(-2, 1, 0)]

∴ (AB)T = [(10, 7, -2),(-6, -5, 1),(4, 6, 0)]       ...(i)

Now,AT = [(2, 3, 0),(-4, -2, 1)] "and B"^"T" = [(1, -2),(-1, 1),(2, 0)]

∴ BTAT = [(1, -2),(-1, 1),(2, 0)] [(2, 3, 0),(-4, -2, 1)]

= [(2 + 8, 3 + 4, 0 - 2),(-2 - 4, -3 - 2, 0 + 1),(4 - 0, 6 - 0, 0 + 0)]

∴ BTAT = [(10, 7, -2),(-6, -5, 1),(4, 6, 0)]         ...(ii)

From (i) and (ii), we get
(AB)T = BTAT.

Concept: Properties of Matrices
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 2 Matrices
Miscellaneous Exercise 2 | Q 4.12 | Page 84