Q.Consider the matrices $A=\begin{bmatrix}4 & 6 & -1\\3 & 0 & 2\\1 & -2 & 5\end{bmatrix}$, $B=\begin{bmatrix}2 & 4\\0 & 1\\-1 & 2\end{bmatrix}$, $C=\begin{bmatrix}3\\1\\2\end{bmatrix}$ out of the given matrix product ............ i) $(AB)^TC$ ii) $C^TC(AB)^T$ iii) $C^TAB$ iv) $A^TABB^TC$ (A) Exactly one is defined (B) Exactly two are defined (C) Exactly three are defined (D) all four are defined

Two matrices $A_{m\times n}$ and $B_{n\times p}$ can be multiplied — "are conformable for the product $AB$" — exactly when the number of columns of $A$ matches the number of rows of $B$; the product $AB$ then has order $m\times p$, with each entry $C_{ij}=\sum_k a_{ik}b_{kj}$ formed by "dotting" row $i$ of $A$ with column $j$ of $B$. Matrix multiplication behaves very differently from ordinary number multiplication: it is generally not commutative ($AB\ne BA$, and one product may not even be defined while the other is); it is associative ($(AB)C=A(BC)$) and distributes over addition ($A(B+C)=AB+AC$); multiplying by the identity matrix $I$ leaves a matrix unchanged ($AI=IA=A$); and — sharply unlike numbers — the product of two non-zero matrices can equal the zero matrix. Because $AB\ne BA$, algebraic identities that are automatic for numbers (like $(A+B)^2=A^2+2AB+B^2$) pick up extra cross-terms for matrices: $(A+B)^2=A^2+AB+BA+B^2$. Matrix multiplication is also how real data — quantities and prices, or students and fees — gets combined in one shot, which is why word problems in this chapter (bookseller totals, sports-academy fees, purchase amounts) are all set up as a single matrix product.