2.2 Q-2

Exercise 2.2 — Question 2

This question practices matrix multiplication with real and/or complex entries, a core operation under SLO M-11-A-15.

Key Rule: Matrix Multiplication

For two matrices $A$ (of order $m\times n$) and $B$ (of order $n\times p$), the product $C=AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. The resulting matrix $C$ has order $m\times p$.

The element in row $i$ and column $j$ of $C$ is: $c_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$

Key Rule: Matrix Addition / Subtraction

Two matrices $A$ and $B$ can be added or subtracted only if they have the same order (same number of rows and columns). The operation is performed element-wise: $(A\pm B{)}_{ij}=a_{ij}\pm b_{ij}$

Worked Example

Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}0& 1\\ 2& 3\end{array}\right)$

Find $AB$:

$AB=\left(\begin{array}{cc}(1)(0)+(2)(2)& (1)(1)+(2)(3)\\ (3)(0)+(4)(2)& (3)(1)+(4)(3)\end{array}\right)=\left(\begin{array}{cc}4& 7\\ 8& 15\end{array}\right)$

Find $A+B$:

$A+B=\left(\begin{array}{cc}1+0& 2+1\\ 3+2& 4+3\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 5& 7\end{array}\right)$

Complex Entries

The same rules apply when entries are complex numbers. For example, if $A=\left(\begin{array}{cc}1+i& 2\end{array}\right)$ and $B=\left(\begin{array}{c}1-i\\ 3\end{array}\right)$, then: $AB=(1+i)(1-i)+(2)(3)=(1-i^2)+6=2+6=8$