Inverse and Transpose of mathematical matrix?

Question

Please show me how to solve the following (step-by-step) thanks.

Use Stata or Excel to find the inverse of 
A= 	
[2	3	1]
[4	5	0]
[2	3	2]

Write out A^T, the transpose of the matrix A. 

Find the inverse of A^T by partitioning 
A^T = 	[B11	B12]	Where B11 is 2x2
	[B21	B22]

Confirm that in this case (A–1)^T =(A^T)–1.

? · Accepted Answer

There are instructions for excel here

http://www.math.iupui.edu/~momran/m118/matrices3.pdf

I would guess that MTRANSPOSE or TRANSPOSE does the transpose in excel.

A^{-1} =

[-5 3/2 5/2]
[4   -1  -2]
[-1   0   1]

A^T =
[2 4 2]
[3 5 3]
[1 0 2]

(A^T)^{-1}=
[-5 4 -1]
[3/2 -1 0]
[5/2 -2 1]

This is indeed (A^{-1})^T from above.

For partitioning, follow the instructions on

http://www.cs.nthu.edu.tw/~jang/book/addenda/matinv/matinv/

where m is 2 and n is 1--there is even a worked example where n = 1.

It's a little tedious but it's doable.

Lauren · Answer

I do not know that there is any special name for the rule. It follows fairly easily from the main multiplication rule for transposes, (AB)T = (B)T(A)T What we want to show is that ((A)-1)T is the inverse of (A)T. Let's check that out. By the general transpose rule, (A)T ((A-1))T = ((A)-1) A)T = (I)T = I, where I is the identity matrix. That tells us that ((A)-1)T is the inverse of (A)T, which is what we wanted to prove.

Inverse and Transpose of mathematical matrix?

2 Answers