SOLUTION: If A is an invertible 2 x 2 matrix,then A^t is invertible and (At)^-1 = (A^-1)^t. is this statement true or false and why?

Question 1006566: If A is an invertible 2 x 2 matrix,then A^t is invertible and (At)^-1 =
(A^-1)^t.
is this statement true or false and why?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Note: the notation "[a,b;c,d]" means "the matrix with the first row being a,b and the second row being c,d"

Matrix A is some 2 x 2 matrix, so let's set it up as
A = [a,b;c,d]
where a,b,c,d are constants

A is invertible ---> determinant D = ad-bc is nonzero

A^T = [a,c;b,d]
determinant of A^T = ad-bc = determinant of matrix A

Note: only b and c swap places when we go from A to A^T. So both A and A^T have the same determinant

Since A has a nonzero determinant, so does A^T.

If A is invertible, then A^T is definitely invertible.

So the first part of the claim has been proven true.

----------------------------------------------------------------

Let's compute the inverse of matrix A

A = [a,b;c,d]
A^(-1) = 1/D * [d,-b;-c,a]

Now let's transpose A^(-1) to get
(A^(-1))^T = 1/D * [d,-c;-b,a]

--------------------------

Now compute the inverse of matrix A^T

A^T = [a,c;b,d]
(A^T)^(-1) = 1/D * [d,-c;-b,a]

--------------------------------
So,

(A^T)^(-1) = 1/D * [d,-c;-b,a]
and
(A^(-1))^T = 1/D * [d,-c;-b,a]

we can see that (A^T)^(-1) = (A^(-1))^T is true.

The second part of the statement has been proven true.

----------------------------------------------------------------

Both parts of the statement are true. A^T is invertible and the equation given holds true.

RELATED QUESTIONS
If the matrices A and B are two invertible matrices, then (A + B) is also invertible... (answered by ikleyn)

If (1 2 1) is a solution of matrix equation Ax=B. Determine and explain why is each... (answered by ikleyn)

Q−1: [5×2 marks] Answer each of the following as True or False justifying your... (answered by math_tutor2020)

True or false questions: 1. In order for a matrix B to be the inverse of A, both... (answered by stanbon)

A square matrix A is called symmetric if A^t = A. If B is a square matrix, then B.B^t and (answered by jim_thompson5910)

suppose A is a 2 x 2 matrix as follows A = [56 34] how do i decide whether A (answered by Fombitz)

Prove that if λ is an eigencalue of an invertible matrix A and x is a corresponding... (answered by narayaba)

Find the values of x, y, and z so that matrix A = 1 2 x 3 0 y 1 1 z is... (answered by Edwin McCravy)

a square matrix A is said to be idempotent A^2=A (a)give an example of idempotent... (answered by Edwin McCravy)