# Hard Lin Alg.

• Nov 5th 2007, 08:30 PM
caeder012
Hard Lin Alg.
Suppose we have two matrices, matrix A and matrix B, both of which are 5x5 matrices. Further, suppose that all the entries in A and B are nonzero. However, the third column in the PRODUCT of AB is all zero.

1.) Can matrix A be invertible? If so, give an example. If not, justify why not.

2.) Can matrix B be invertible? If so, give an example. If not, justify why not.

We were given this hint:

"Consider what the definition of the product AB of 2 matrices means"
• Nov 6th 2007, 02:35 AM
Plato
Quote:

Originally Posted by caeder012
Suppose we have two matrices, matrix A and matrix B, both of which are 5x5 matrices.
Further, suppose that all the entries in A and B are nonzero. However, the third column in the PRODUCT of AB is all zero.
Can matrix A be invertible? If so, give an example. If not, justify why not.

If A were invertible then $\displaystyle A^{ - 1} \left( {AB} \right) = ?$
Would that matrix necessarily have zero entries?