Exercise 2.1 — Question 2

Key Concepts

This question applies matrix addition, subtraction, and multiplication with real and complex entries.

Condition for Addition / Subtraction

Two matrices $A$ and $B$ can be added or subtracted only if they have the same order ($m\times n$). The result is computed element-wise:

$(A\pm B{)}_{ij}=a_{ij}\pm b_{ij}$

Condition for Multiplication

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. If $A$ is of order $m\times n$ and $B$ is of order $n\times p$, then:

$(AB{)}_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$

and the resulting matrix $AB$ has order $m\times p$.

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Worked Examples

Example 1 — Matrix Addition with Complex Entries

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

Since both matrices have order $2\times 2$, addition is defined:

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

Example 2 — Matrix Subtraction

Using the same matrices:

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

Example 3 — Matrix Multiplication

Let $A={\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)}_{2\times 3},B={\left(\begin{array}{cc}7& 8\\ 9& 10\\ 11& 12\end{array}\right)}_{3\times 2}$

Since the inner dimensions match ($3=3$), $AB$ is defined with order $2\times 2$:

$AB=\left(\begin{array}{cc}1(7)+2(9)+3(11)& 1(8)+2(10)+3(12)\\ 4(7)+5(9)+6(11)& 4(8)+5(10)+6(12)\end{array}\right)=\left(\begin{array}{cc}58& 64\\ 139& 154\end{array}\right)$

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