Properties of Transpose of the Matrices

The value of b for which the function f(x) = x + cos x + b is strictly decreasing over R is:

Transpose of a column matrix is a column matrix.

If A = `[(costheta, sintheta),(-sintheta, costheta)]`, then show that A² = `[(cos2theta, sin2theta),(-sin2theta, cos2theta)]`

Amit, Biraj, and Chirag were given the task of creating a square matrix of order 2.
Below are the matrices created by them. A, B, C are the matrices created by Amit, Biraj, and Chirag respectively.

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

If a = 4 and b = -2, based on the above information answer the following:

• `("bA")^"T"` is equal to ____________.

Show that A = `[(5, 3),(-1, -2)]` satisfies the equation A² – 3A – 7I = O and hence find A^–1.

If: `[(2, 1, 3)] [(-1, 0, -1),(-1, 1, 0),(0, 1, 1)] [(1),(0),(-1)]` = A, find A

Amit, Biraj, and Chirag were given the task of creating a square matrix of order 2.
Below are the matrices created by them. A, B, C are the matrices created by Amit, Biraj, and Chirag respectively.

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

If a = 4 and b = -2, based on the above information answer the following:

• `("A"^"T")^"T"` is equal to ____________.

Amit, Biraj, and Chirag were given the task of creating a square matrix of order 2.
Below are the matrices created by them. A, B, C are the matrices created by Amit, Biraj, and Chirag respectively.

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

If a = 4 and b = -2, based on the above information answer the following:

• (a + b)B is equal to ____________.