2.2 Q-6

Exercise 2.2 — Question 6

Topic: Matrix Multiplication

This question involves performing matrix multiplication, applying SLO M-11-A-15: matrix operations with real and complex entries.

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Key Concepts

Condition for Multiplication: Two matrices $A$ and $B$ can be multiplied as $AB$ only if 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 $AB$ is of order $m\times p$.

Formula for each element: $[AB{]}_{ij}={\sum }_{k=1}^n{a}_{ik}\cdot b_{kj}$

That is, the $(i,j)$ entry of $AB$ is the dot product of the $i$-th row of $A$ with the $j$-th column of $B$.

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Important Properties

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Worked Example

Let: $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)$

Find $AB$:

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

$AB=\left(\begin{array}{cc}5+14& 6+16\\ 15+28& 18+32\end{array}\right)=\left(\begin{array}{cc}19& 22\\ 43& 50\end{array}\right)$

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Example with Complex Entries

Let: $A=\left(\begin{array}{cc}1+i& 2\\ 0& 1-i\end{array}\right),B=\left(\begin{array}{c}1\\ i\end{array}\right)$

Find $AB$:

$AB=\left(\begin{array}{c}(1+i)(1)+(2)(i)\\ (0)(1)+(1-i)(i)\end{array}\right)=\left(\begin{array}{c}1+i+2i\\ i-i^2\end{array}\right)=\left(\begin{array}{c}1+3i\\ i+1\end{array}\right)$

(using $i^2=-1$)

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Practice

Attempt the specific matrices given in Q6 of Exercise 2.2 in your textbook using the row-by-column method above.