Matrix 2x1 transpose - inverse

Question

If A and B are any matrices of order 2×1, why does the product ABt have no inverse?

Answer 1

AB^t  for A=[a] and B= [c]

                [b]             [d]

AB^t=[a][c d]  = [ac   ad]

       [b]            [bc   bd] 

The determinant is clearly zero.

I wish there were a better way to display matrices.

Answer 2

Hi -

 

I read the problem a little differently -- I read "ABt" to be A * Bt -- that is, matrix A multiplied by the transpose of matrix B. Unlike the other answer (which is certainly a valid answer if you read the problem as A * B, then transpose), this answer does give a proper multiplication.

 

Let's try an example: matrix A is

[2]

[5]

 

and B is

[9]

[7]

 

Both are 2 rows x 1 column. The transpose of B is Bt=[9 7], a 1 row x 2 column matrix. The product of A and Bt is

 

[2] * [9 7]

[5]

 

= [2*9  2*7]

   [5*9  5*7]

 

= [18  14]  = A*Bt

   [45  35]

 

Taking the inverse of A*Bt, we get

 

[35  -14]  * 1 / (18*35 - 14*45)

[-45  18]

 

with (18*35 - 14*45) being D, the "determinate". As you can see, D=0. Division by zero is not allowed, so that is why there NO INVERSE !

 

I leave it to you to work backwards from D to the constituent values, and convince yourself that D=0 is always true for any A and B matricies.

 

Best wishes, Pete

Answer 3

A matrix can be multiplied by another matrix only if the first matrix has the same number of columns as the second one of rows.

The multiplication can not be performed and then there is no inverse to the nonexistent product AB

I understand that the two matrices are any 2x1 matrices. I do not see numbers that may or may not give a determinant equal to zero.