Final answer:
To prove that matrix 'a' is invertible if and only if matrices 'b' and 'c' are invertible, one must show the implications in both directions using properties of matrix multiplication and inverses.
Step-by-step explanation:
The question is asking to prove that a matrix 'a' is invertible if and only if both matrices 'b' and 'c' are invertible. Assuming there is a relationship between matrices a, b, and c such as a product or sum, one would need to show two implications: first, if 'a' is invertible then 'b' and 'c' are invertible; second, if 'b' and 'c' are invertible then 'a' is also invertible. In general, this involves using properties of matrix multiplication, such as associativity, distributivity, and the existence of multiplicative inverses for invertible matrices.
To establish this proof, you would typically assume that 'a' is invertible and show that this implies 'b' and 'c' must be invertible, perhaps by constructing a scenario where 'b' and 'c' are multiplied by 'a' or its inverse and properties of invertible matrices apply. Conversely, you would assume that 'b' and 'c' are invertible, and then demonstrate that 'a' must also be invertible, again by using the properties of matrix operations and inverses.