Question
Consider the matrices \[A = \left[ {\begin{array}{*{20}{c}}
4&6&{ - 1}\\
3&0&2\\
1&{ - 2}&5
\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}
2&4\\
0&1\\
{ - 1}&2
\end{array}} \right],C = \left[ {\begin{array}{*{20}{c}}
3\\
1\\
2
\end{array}} \right]\] Out of the given matrix products, which one is not defined
A.
$${\left( {AB} \right)^T}C$$
B.
$${C^T}C{\left( {AB} \right)^T}$$
C.
$${C^T}AB$$
D.
$${A^T}AB{B^T}C$$
Answer :
$${C^T}C{\left( {AB} \right)^T}$$
Solution :
$$\eqalign{
& A \to 3 \times 3,B \to 3 \times 2,C \to 3 \times 1 \cr
& AB \to 3 \times 2 \cr
& \Rightarrow \,{\left( {AB} \right)^T} = 2 \times 3 \cr
& \Rightarrow \,{\left( {AB} \right)^T}C\,{\text{is}}\,{\text{defined}} \cr
& \Rightarrow \,{C^T} \to 1 \times 3, \cr
& \Rightarrow \,{C^T}C \to 1 \times 1 \cr} $$
Hence, $${C^T}C{\left( {AB} \right)^T}$$ is not defined. Now, $$C^T AB$$ is also defined.
$$\eqalign{
& {A^T} \to 3 \times 3,{B^T} \to 2 \times 3;\,\,\,{A^T}A \to 3 \times 3 \cr
& B{B^T} \to 3 \times 3 \cr
& \Rightarrow \,{A^T}AB{B^T} \to 3 \times 3 \cr
& \Rightarrow \,{A^T}AB{B^T}C\,{\text{is}}\,{\text{defined}} \cr} $$