Question

Compute the indicated products
(i) $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

(ii) $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$

(iii) $\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$

(iv) $\left[\begin{array}{c}2\\ 3\\ 4\end{array}\begin{array}{c}3\\ 4\\ 5\end{array}\begin{array}{c}4\\ 5\\ 6\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 3\end{array}\begin{array}{c}-3\\ 2\\ 0\end{array}\begin{array}{c}5\\ 4\\ 5\end{array}\right]$

(v) $\left[\begin{array}{c}2\\ 3\\ -1\end{array}\begin{array}{c}1\\ 2\\ 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 1\\ -1& 2& 1\end{array}\right]$

(vi) $\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]\left[\begin{array}{c}2\\ 1\\ 3\end{array}\begin{array}{c}-3\\ 0\\ 1\end{array}\right]$

Answer 1

This is a matrix product

$A$ and $B$ are two matrix and $AB$ represents their product

(i)

Here

$A=\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$

$B=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

Here we multiply first row of $A$ with first column of $B$ and that will make first element for the first row of product matrix.

i.e First element of $AB$ will be $a\ast a+b\ast b=a^2+b^2$

We will get second element of first row for product matrix by multiplying first row of $A$ with second column of $B$

i.e $-a\ast b+b\ast a=0$

We will get first element of second row for product matrix by multiplying second row of $A$ with first column of $B$

i.e $-b\ast a+a\ast b=0$

We will get second element of second row for product matrix by multiplying second row of $A$ with second column of $B$

i.e $-b\ast -b+a\ast a=-b^2+a^2$

So final product matrix $AB$ will be

$AB=\left[\begin{array}{cc}{a}^2+b^2& 0\\ 0& a^2+b^2\end{array}\right]$

(ii)

Here

$A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

$B=\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$

Here we multiply first row of $A$ with first column of $B$ and that will make first element for the first row of product matrix.

i.e First element of $AB$ will be $1\ast 2=2$

We will get second element of first row for product matrix by multiplying first row of $A$ with second column of $B$

i.e $1\ast 3=3$

We will get third element of first row for product matrix by multiplying first row of $A$ with third column of $B$

i.e $1\ast 4=4$

We will get first element of second row for product matrix by multiplying second row of $A$ with first column of $B$

i.e $2\ast 2=4$

We will get second element of second row for product matrix by multiplying second row of $A$ with second column of $B$

i.e $2\ast 3=6$

We will get third element of second row for product matrix by multiplying second row of $A$ with third column of $B$

i.e $2\ast 4=8$

We will get first element of third row for product matrix by multiplying third row of $A$ with first column of $B$

i.e $3\ast 2=6$

We will get second element of third row for product matrix by multiplying third row of $A$ with second column of $B$

i.e $3\ast 3=9$

We will get third element of third row for product matrix by multiplying third row of $A$ with third column of $B$

i.e $3\ast 4=12$

So final product matrix $AB$ will be

$AB=\left[\begin{array}{ccc}2& 3& 4\\ 4& 6& 8\\ 6& 9& 12\end{array}\right]$

(iii)

$A=\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]$

$B=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$

$AB=$ $\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]$x$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$

$\Rightarrow$ $AB=\left[\begin{array}{ccc}1\ast 1+(-2\ast 2)& 1\ast 2+(-2\ast 3)& 1\ast 3+(-2\ast 1)\\ 2\ast 1+3\ast 2& 2\ast 2+2\ast 3& 2\ast 3+3\ast 1\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{ccc}-3& -4& 1\\ 8& 13& 9\end{array}\right]$

(iv)

$A=\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]$

$B=\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right]$

$AB=$ $\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]$x$\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right]$

$\Rightarrow$ $AB=\left[\begin{array}{ccc}2\ast 1+3\ast 0+4\ast 3& (2\ast -3)+3\ast 2+4\ast 0& 2\ast 5+3\ast 4+4\ast 5\\ 3\ast 1+4\ast 0+5\ast 3& (3\ast -3)+4\ast 2+5\ast 0& 3\ast 5+4\ast 4+5\ast 5\\ 4\ast 1+5\ast 0+6\ast 3& (4\ast -3)+5\ast 2+6\ast 0& 4\ast 5+5\ast 4+6\ast 5\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{ccc}15& 0& 42\\ 18& -1& 56\\ 22& 2& 70\end{array}\right]$

(v)

$A=\left[\begin{array}{cc}2& 1\\ 3& 2\\ -1& 1\end{array}\right]$

$B=\left[\begin{array}{ccc}1& 0& 1\\ -1& 2& 1\end{array}\right]$

$\Rightarrow AB=$ $\left[\begin{array}{cc}2& 1\\ 3& 2\\ -1& 1\end{array}\right]$x$\left[\begin{array}{ccc}1& 0& 1\\ -1& 2& 1\end{array}\right]$

$\Rightarrow$ $AB=\left[\begin{array}{ccc}2\ast 1+(1\ast -1)& 2\ast 0+1\ast 2& 2\ast 1+1\ast 1\\ 3\ast 1+(2\ast -1)& 3\ast 0+2\ast 2& 3\ast 1+2\ast 1\\ (-1\ast 1)+(1\ast -1)& (-1\ast 0)+1\ast 2& (-1\ast 1)+1\ast 1\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{ccc}1& 2& 3\\ 1& 4& 5\\ -2& 2& 0\end{array}\right]$

(vi)

$A=\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]$

$B=\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]$

$\Rightarrow AB=$$\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]$x$\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]$

$\Rightarrow$ $AB=\left[\begin{array}{cc}3\ast 2+(-1\ast 1)+3\ast 3& (3\ast -3)+-1\ast 0+3\ast 1\\ (-1\ast 2)+0\ast 1+2\ast 3& (-1\ast -3)+(0\ast 0)+2\ast 1\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{cc}14& 0\\ 4& 5\end{array}\right]$