Matrix Multiplication - Practice Questions & MCQ

Matrix multiplication: 

Product AB can be found if the number of columns in matrix A and the number of rows in matrix B are equal. Otherwise, multiplication AB is not possible.

i) AB is defined only if col(A) = row(B)

ii) BA is defined only if col(B) = row(A)

If 

$
\begin{aligned}
& \mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{I}} \\
& B=\left[b_{i j}\right]_{n \times \mathrm{P}} \\
& \mathrm{C}=\mathrm{AB}=\left[\mathrm{c}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{p}} \\
& \text { Where } c_{\mathrm{ij}}=\sum_{\mathrm{j}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ijj}} \mathrm{~b}_{\mathrm{jk}}, 1 \leq \mathrm{i} \leq \mathrm{m}, 1 \leq \mathrm{k} \leq \mathrm{p} \\
& =a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+a_{i 3} b_{3 k}+\ldots+a_{i n} b_{n k}
\end{aligned}
$
For examples

Suppose, two matrices are given

$
\mathrm{A}=\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{33}
\end{array}\right]_{2 \times 3} \text { and } \mathrm{B}=\left[\begin{array}{lll}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array}\right]_{3 \times 3}
$
To obtain the entries in row $i$ and columnj of AB , we multiply the entries in row $i$ of A by column $j$ in B and add.
given matrices A and B , where the order of A are $2 \times 3$ and the order of B are $3 \times 3$, the product of AB will be a $2 \times 3$ matrix.

To obtain the entry in row 1 , column 1 of AB , multiply the first row in A by the first column in B , and add.

$
\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13}
\end{array}\right]\left[\begin{array}{l}
b_{11} \\
b_{21} \\
b_{31}
\end{array}\right]=\mathrm{a}_{11} \cdot \mathrm{~b}_{11}+\mathrm{a}_{12} \cdot \mathrm{~b}_{21}+\mathrm{a}_{13} \cdot \mathrm{~b}_{31}
$

To obtain the entry in row 1 , column 2 of AB , multiply the first row in A by the second column in B , and add.

$
\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13}
\end{array}\right]\left[\begin{array}{l}
b_{12} \\
b_{22} \\
b_{32}
\end{array}\right]=\mathrm{a}_{11} \cdot \mathrm{~b}_{12}+\mathrm{a}_{12} \cdot \mathrm{~b}_{22}+\mathrm{a}_{13} \cdot \mathrm{~b}_{32}
$
To obtain the entry in row 1 , column 3 of AB , multiply the first row in A by the third column in B , and add.

$
\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13}
\end{array}\right]\left[\begin{array}{l}
b_{13} \\
b_{23} \\
b_{33}
\end{array}\right]=\mathrm{a}_{11} \cdot \mathrm{~b}_{13}+\mathrm{a}_{12} \cdot \mathrm{~b}_{23}+\mathrm{a}_{13} \cdot \mathrm{~b}_{33}
$
We proceed the same way to obtain the second row of AB . In other words, row 2 of A times column 1 of $B$; row 2 of $A$ times column 2 of $B$; row 2 of A times column 3 of B.
When complete, the product matrix will be

$
\mathrm{AB}=\left[\begin{array}{lll}
a_{11} \cdot b_{11}+a_{12} \cdot b_{21}+a_{13} \cdot b_{31} & a_{11} \cdot b_{12}+a_{12} \cdot b_{22}+a_{13} \cdot b_{32} & a_{11} \cdot b_{13}+a_{12} \cdot b_{23}+a_{13} \cdot b_{33} \\
a_{21} \cdot b_{11}+a_{22} \cdot b_{21}+a_{23} \cdot b_{31} & a_{21} \cdot b_{12}+a_{22} \cdot b_{22}+a_{23} \cdot b_{32} & a_{21} \cdot b_{13}+a_{22} \cdot b_{23}+a_{23} \cdot b_{33}
\end{array}\right]
$
 

Properties of matrix multiplication:

i) Multiplication may or may not be commutative, so AB may or may not be equal to BA .
ii) Matrix multiplication is associative, meaning $A(B C)=(A B) C$
iii) Matrix multiplication is distributive over addition, mean $A(B+C)=A B+A C$ and $(B+C) A=B A+C A$
iv) If matrix multiplication of two matrices gives a null matrix then it doesn't mean that any of those two matrices was a null matrix.

So $A B=O \nRightarrow A=O$ or $B=O$.

$A=\left[\begin{array}{ll}0 & 2 \\ 0 & 0\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$, then $A B=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

v) Cancellation law in matrix multiplication doesn't hold, which means $A B=A C \nRightarrow B=C$

 vii) if A is m x n matrix then, $\mathrm{I}_{\mathrm{m}} \mathrm{A}=\mathrm{A}=\mathrm{AI}_{\mathrm{n}}$.