Exercise 2.1 — Question 4

Problem Statement

Perform the indicated matrix operations. (Typical Q4 problems involve addition, subtraction, and/or multiplication of matrices with real or complex entries.)

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Key Concepts Required

Matrix Addition and 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:

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

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

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

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

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Matrix Multiplication

Matrix $A$ of order $m\times n$ can be multiplied by matrix $B$ of order $n\times p$ (inner dimensions must match). The product $AB$ has order $m\times p$, and its $(i,j)$-th element is:

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

Example: If $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ (order $2\times 2$) and $B=\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)$ (order $2\times 2$), then:

$AB=\left(\begin{array}{cc}1\cdot 5+2\cdot 7& 1\cdot 6+2\cdot 8\\ 3\cdot 5+4\cdot 7& 3\cdot 6+4\cdot 8\end{array}\right)=\left(\begin{array}{cc}19& 22\\ 43& 50\end{array}\right)$

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Operations with Complex Entries

The same rules apply when matrix entries are complex numbers. Apply standard complex arithmetic ($i^2=-1$) when computing each element.

Example: If $A=\left(\begin{array}{cc}i& 1\end{array}\right)$ and $B=\left(\begin{array}{c}i\\ 2\end{array}\right)$, then:

$AB=i\cdot i+1\cdot 2=i^2+2=-1+2=1$

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Worked Steps (General Approach)

1. Identify the order of each matrix involved.
2. Check compatibility: same order for addition/subtraction; inner dimensions match for multiplication.
3. Compute element-wise for addition/subtraction, or use the dot-product row-by-column rule for multiplication.
4. Simplify any complex number arithmetic ($i^2=-1$, combine real and imaginary parts).

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