Question

We are told that a certain$\mathbf{5}\mathbf{\times }\mathbf{5}$ matrix$\mathit{A}$ can be written as

$\mathit{A}\mathbf{=}\mathit{B}\mathit{C}$,

where $\mathit{B}$is $\mathbf{5}\mathbf{\times }\mathbf{4}$and$\mathit{C}$ is $\mathbf{4}\mathbf{\times }\mathbf{5}$. Explain how you know that$\mathit{A}$ is not invertible.

Short answer

Thus, it is proved matrix$A$ is not invertible.

Step-by-step solution

Given in the question.

Let matrix $A$is $5\times 5$which is $A=BC$where$B$ is $5\times 4$and$C$ is$4\times 5$ .

Let assume that$colspace\left(A\right)\subseteq colspace\left(B\right)$ . There are different ways and one way is as the linear transformation$T_A$ is the composition of $T_B$and$T_C$ gives $im\left(T_A\right)\subseteq im\left(T_B\right)$.

Explanation of A is not invertible.

Take dimensions where$rank\left(A\right)\le rank\left(B\right)$. Now by the rank-nullity theorem applied to $B$, gives$rank\left(B\right)\le 4$.

Which implies that$rank\left(A\right)$is never 5, that is needed for $A$to be invertible.

Hence, matrix$A$is not invertible.