[3 marks sum]

Question 13 Marks

Let $A$ be a $2 \times 2$ matrix and let $I$ be an identity matrix of the order $2 \times 2.$ Prove that $AI = IA = A.$

Answer

$\begin{aligned} & \text { Let } A=\left|\begin{array}{ll}p & q \\ r & s\end{array}\right|_{2 \times 2} \end{aligned} $
$ A I=\left|\begin{array}{ll}p & q \\ r & s\end{array}\right|\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right| $
$ =\left|\begin{array}{ll}p+0 & 0+q \\ r+0 & 0+s\end{array}\right| $
$ =\left|\begin{array}{ll}p & q \\ r & s\end{array}\right|=A $
$ I A=\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\left|\begin{array}{ll}p & q \\ r & s\end{array}\right| $
$ =\left|\begin{array}{ll}p+0 & q+0 \\ 0+r & 0+s\end{array}\right| $
$ =\left|\begin{array}{ll}p & q \\ r & s\end{array}\right|=a$
Hence proved $A|=| A=A$.

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Question 23 Marks

Evaluate the following $: \left|\begin{array}{ll} \mid 3 & 2 \end{array}\right|\left|\begin{array}{c} -1 \\ 3 \end{array}\right| $

Answer

$\begin{array}{l}\left|\begin{array}{cc|c}\mid 3 & \left.2\right|_{1 \times 2} & -1 \\ 3\end{array}\right|_{2 \times 1} \end{array} $
$ =|3 \times-1+2 \times 3| $
$ =|-3+6| $
$ =|3|_{1 \times 1} $

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Question 33 Marks

If $A =\left|\begin{array}{l}1215 \\ 1117\end{array}\right|$ and $B =\left|\begin{array}{ll}2 & 7 \\ 4 & 9\end{array}\right|$ find $:2 A +3 B$

Answer

$\begin{array}{l}A=\left|\begin{array}{ll}12 & 15 \\ 11 & 17\end{array}\right|_{2 \times 2}, B=\left|\begin{array}{ll}2 & 7 \\ 4 & 9\end{array}\right|_{2 \times 2}\end{array} $
$ 2 A+3 B $
$ 2 A=\left|\begin{array}{ll}24 & 30 \\ 22 & 34\end{array}\right|, 3 B=\left|\begin{array}{cc}6 & 21 \\ 12 & 27\end{array}\right| $
$ 2 A+3 B=\left|\begin{array}{cc}24+6 & 30+21 \\ 22+12 & 34+27\end{array}\right| $
$ =\left|\begin{array}{ll}30 & 51 \\ 34 & 61\end{array}\right|_{2 \times 2}$

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Question 43 Marks

Find the values of $x$ and $y$, if $\left|\begin{array}{c}3 x-y \\ 5\end{array}\right|=\left|\begin{array}{c}7 \\ x+y\end{array}\right|$

Answer

$\left|\begin{array}{c} 3 x-y \\ 5 \end{array}\right|_{2 \times 1}$
$=\left|\begin{array}{c} 7 \\ x+y \end{array}\right|_{2 \times 1} $
$ 3 x-y=7-(1) $
$ x+y=5-(2) $
$\Rightarrow x=5-y$
Putting the value of $x$ in $(1)$
$3(5-y)-y=7$
$\Rightarrow 15-3 y-y=7$
$\Rightarrow-4 y=-8$
$\Rightarrow y=2$
from $(2)$
$x+2=5$
$\Rightarrow x =3$

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Question 53 Marks

Find the values of a and b) if [2a + 3b a - b] = [19 2].

Answer

[2a+3b a-b] = [19 2]
2a+3b a-b] is 1 x 2 matrix and [19 2] is 1 x 2 matrix
2a + 3b = 19 ......(1)
a - b = 2 ......(2)
⇒ a = 2 + b
Putting the value of a in equation (1)
4 + 2b + 3b = 19 .... (1)
⇒ 5b= 15
⇒ b = 3
From (2)
a = 2+3 = 5

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Question 63 Marks

If $P =\left|\begin{array}{cc}14 & 17 \\ 13 & 1\end{array}\right|$ and $Q =\left|\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right|,$ find matrix $M$ such that $P - M =3 Q$

Answer

$\begin{array}{l}P=\left|\begin{array}{cc}14 & 17 \\ 13 & 1\end{array}\right|_{2 \times 2}\end{array} $
$Q=\left|\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right|_{2 \times 2} $
$ P-M=3 Q $
$ M=P-3 Q $
$ M=\left|\begin{array}{cc}14 & 17 \\ 13 & 1\end{array}\right|-\left|\begin{array}{cc}6 & 3 \\ 9 & -9\end{array}\right| $
$ =\left|\begin{array}{cc}14-6 & 17-3 \\ 13-9 & 1+9\end{array}\right| $
$ =\left|\begin{array}{ll}8 & 14 \\ 4 & 10\end{array}\right|_{2 \times 2}$

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Question 73 Marks

If $A=\left|\begin{array}{ll}15 & 7 \\ 13 & 8\end{array}\right|$ and $B=\left|\begin{array}{ll}16 & 12 \\ 27 & 11\end{array}\right|$, find matrix $X$ such that $2 A-X=B$.

Answer

$\begin{array}{l}A=\left|\begin{array}{ll}15 & 7 \\ 13 & 8\end{array}\right|_{2 \times 2}\end{array}, $
$B=\left|\begin{array}{ll}16 & 12 \\ 27 & 11\end{array}\right|_{2 \times 2} $
$ 2 A-X=B $
$ X=2 A-B $
$ X=\left|\begin{array}{ll}30 & 14 \\ 26 & 16\end{array}\right|-\left|\begin{array}{ll}16 & 12 \\ 27 & 11\end{array}\right| $
$ =\left|\begin{array}{ll}14 & 2 \\ -1 & 5\end{array}\right|_{2 \times 2}$

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Question 83 Marks

If $A =\left|\begin{array}{ll}15 & 7 \\ 13 & 8\end{array}\right|$ and $B =\left|\begin{array}{ll}16 & 12 \\ 27 & 11\end{array}\right|$, find matrix $X$ such that $A + X$

Answer

$\begin{array}{l}A=\left|\begin{array}{ll}15 & 7 \\ 13 & 8\end{array}\right|_{2 \times 2} \end{array}$
$B=\left|\begin{array}{ll}16 & 12 \\ 27 & 11\end{array}\right|_{2 \times 2} $
$ A+X=B $
$ X=B-A $
$ \therefore X=\left|\begin{array}{ll}16 & 12 \\ 27 & 11\end{array}\right|-\left|\begin{array}{ll}15 & 7 \\ 13 & 8\end{array}\right|$
$=\left|\begin{array}{cc}1 & 5 \\ 14 & 3\end{array}\right|_{2 \times 2}$

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Mathematics [3 marks sum]

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