Show that AB=BA where, A = [cosθsinθsinθcosθ],B[cosθ-sinθsinθcosθ] - Mathematics and Statistics

Show that AB = BA where,

A = `[(costheta, - sintheta),(sintheta, costheta)], "B" = [(cosphi, -sinphi),(sinphi, cosphi)]`

Sum

AB = `[(costheta, -sintheta),(sintheta, costheta)] [(cosphi, -sinphi),(sinphi, cosphi)]`

`= [(costhetacosphi - sintheta sinphi, -costhetasinphi - sinthetacosphi),(sinthetacosphi + costhetasinphi, -sinthetasinphi + costhetacosphi)]`

`= [(cos (theta + phi) - sin (theta + phi)), (sin (theta + phi) cos (theta + phi))]` ...(i)

BA = `[(cosphi, -sinphi),(sinphi, cosphi)] [(costheta, - sintheta),(sintheta, costheta)]`

`= [(costhetacosphi - sintheta sinphi, -sinthetacosphi - costhetasinphi),(costhetasinphi + sinthetacosphi, -sinthetasinphi + costhetacosphi)]`

`= [(cos (theta + phi) - sin (theta + phi)), (sin (theta + phi) cos (theta + phi))]` ...(ii)

From (i) and (ii), we get

AB = BA

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Matrices - Properties of Matrix Multiplication

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Chapter 4: Determinants and Matrices - Exercise 4.6 [Page 94]

Chapter 4 Determinants and Matrices
Exercise 4.6 | Q 4. (ii) | Page 94

If A = `[(1, -3),(4, 2)], "B" = [(4, 1),(3, -2)]` show that AB ≠ BA.

If A = `[(-1, 1, 1),(2, 3, 0),(1, -3, 1)], "B" = [(2, 1, 4),(3, 0, 2),(1, 2, 1)]`. State whether AB = BA? Justify your answer.

Verify A(BC) = (AB)C of the following case:

A = `[(1, 0, 1),(2, 3, 0),(0, 4, 5)], "B" = [(2, -2),(-1, 1),(0, 3)] and "C" = [(3, 2, -1),(2, 0, -2)]`

Verify A(BC) = (AB)C of the following case:

A = `[(2, 4, 3),(-1, 3, 2)], "B" = [(2, -2),(3, 3),(-1, 1)], "C" = [(3, 1),(1, 3)]`

Verify that A(B + C) = AB + AC of the following matrix:

A = `[(1, -1, 3),(2, 3, 2)], "B" = [(1, 0),(-2, 3),(4, 3)], "C" = [(1, 2),(-2, 0),(4, -3)]`

If A = `[(1, -2),(5, 6)], "B" = [(3, -1),(3, 7)]`, Find AB - 2I, where I is unit matrix of order 2.

A = `[(alpha, 0),(1, 1)], "B" = [(1, 0),(2, 1)]` find α, if A² = B.

If A = `[(3, 4),(-4, 3)] and "B" = [(2, 1),(-1, 2)]`, show that (A + B)(A – B) = A² – B²

Find k, if A= `[(3, -2),(4, -2)]` and if A² = kA – 2I

Find x and y, if `{4[(2, -1, 3),(1, 0, 2)] -[(3, -3, 4),(2, 1, 1)]} [(2),(-1),(1)] = [(x),(y)]`

Find x, y, z if `{3[(2, 0),(0, 2),(2, 2)] -4[(1, 1),(-1, 2),(3, 1)]} [(1),(2)]` = `[("x" - 3),("y" - 1),(2"z")]`

If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, show that `"A"^2=[(cos2alpha, sin2alpha),(-sin2alpha, cos2alpha)]`

If A = `[(1, 2),(3, 5)]` B = `[(0, 4),(2, -1)]`, show that AB ≠ BA, but |AB| = IAl·IBI

Select the correct option from the given alternatives:

If `[(5, 7),(x, 1),(2, 6)] - [(1, 2),(-3, 5),(2, y)] = [(4, 5),(4, -4),(0, 4)]` then __________

Select the correct option from the given alternatives:

If A = `[(-2, 1),(0, 3)]` and f(x) = 2x² – 3x, then f(A) = ………