Hi all,

I have a matrices equation
A=B.C+D.E (1)
where the nonzero matrices B and D are orthogonal, and have a relationship
B^-1.D=0 (2)
From (1), if we take out B, we have
A=B(C+B^-1.D.E) (3)
And because of (2), (3) becomes
A=B(C+0)=B.C (4)

So (4) is totally different to (1). Where is my error?

Many tks to all responses!

So (4) is totally different to (1). Where is my error?
Your error is in saying that they are different.

In general, two non-zero matrices can have a product that is the zero matrix.

However...

Since the inverse of B exists, and you are given that

B^(-1) . D = 0,

(Where I have used ^(-1) to indicate the inverse of the matrix and the dot "." is used to indicate matrix multiplication.)

Then

B . (B^(-1) . D) = B . 0 = 0

So, since B . B^(-1) is equal to the unit matrix and since matrix multiplication is associative we have

I . D = 0, Where I is the unit (identity) matrix

So it is seen that it must be true that

D = 0

Conclusion: The conditions that you are given imply that D = 0, and your statement that, "(4) is totally different to (1)," is false.

Dave