Question

Kindly answer with a Explanation :-)

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Answer 1

$\bigstar\huge\boxed{\mathtt{\red{Solution:}}}$

Here,

$\implies B=\left[\begin{array}{cc}-4&2\\5&-1\\\end{array}\right]$

$\implies C=\left[\begin{array}{cc}17&-1\\47&-13\\\end{array}\right]$

As per the given condition ,

AB=C

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we know that,

$\implies AB=\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \times\left[\begin{array}{cc}e&f\\g&h\end{array}\right]$

$\implies AB=\left[\begin{array}{cc}ae+gb&af+bh\\ce+dg&cf+dh\end{array}\right]$

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Similarly,

$\implies AB=\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \times\left[\begin{array}{cc}-4&2\\5&-1\end{array}\right]$

$\implies AB=\left[\begin{array}{cc}-4a+5b&2a-b\\-4c+5d&2c-d\end{array}\right]$

$\implies \left[\begin{array}{cc}-4a+5b&2a-b\\-4c+5d&2c-d\end{array}\right]=\left[\begin{array}{cc}17&-1\\47&-13\end{array}\right]$

From this we can conclude that ,

$\rightarrow -4a+5b=17 ....(1)$

$\rightarrow 2a -b=-1 ....(2)$

Multiplying (2) by 2 we get ,

$\rightarrow 4a -2b=-2 ....(3)$

Adding equations (1) and (3) we get ,

$\rightarrow 3b= 15$

$\rightarrow \boxed{b=5 }$

Now substituting the value of b in (2) we get ,

$\rightarrow 2a -5=-1$

$\rightarrow 2a = 4$

$\rightarrow \boxed{a = 2}$

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Similarly,

$\rightarrow -4c+5d=47 ....(4)$

$\rightarrow 2c -d=-13 ....(5)$

Multiplying (5) by 2 we get ,

$\rightarrow 4c -2d=-26 ....(6)$

Adding equations (4) and (6) we get ,

$\rightarrow 3d= 21$

$\rightarrow \boxed{d=7}$

Now substituting the value of d in (5) we get ,

$\rightarrow 2c -7=-13$

$\rightarrow 2c =-6$

$\rightarrow \boxed{c =-3}$

$\implies A=\pink{\left[\begin{array}{cc}2&5\\-3&7\\\end{array}\right]}$

Answer 2

${\huge{\fcolorbox{purple}{pink}{\fcolorbox{yellow}{red}{\bf{\color{white}{ǫᴜᴇsᴛɪᴏɴ}}}}}}$

$If B = \binom{ - 4 \: \: \: \: 2}{5 \: \: \: \: - 1}and \: c = \binom{17 \: \: \: \: - 1}{47 \: \: \: \: - 13} find \: matrix \: a \: such \: that \: ab = c$

${\huge{\fcolorbox{purple}{pink}{\fcolorbox{yellow}{red}{\bf{\color{white}{ᴀɴsᴡᴇʀ}}}}}}$

Find the matrix A such that AB = C

$B = \binom{ - 4 \: \: \: \: 2}{5 \: \: \: \: - 1} \\ c = \binom{17 \: \: \: \: - 1}{47 \: \: \: \: - 13} \\ and \: ab = c \\ let \: a = \binom{a \: \: \: \: b}{c \: \: \: \: d} \\ then \: ab = \binom{a \: \: \: \: b}{c \: \: \: \: d} \\ \times { - 4 \: \: \: \: 2}{5 \: \: \: \: - 1} \\ = \binom{ - 4a + 5b \: 2a - b}{ - 4c + 5d \: 2c - d}$ \\ AB = C \\ \binom{ - 4a

Comparing corresponding elements , we get

=> -4a + 5b = 17

=> 2a - b = -1

=> -4c + 5d = 47

=> 2c - d = -13

multiplying by 1 and by 2

==> -4a + 5b = 17

4a - 2b = -2

Adding

3b = 15

=>b = 15/3 = 5

=>2a - b = -1

=> 2a = -1 + 5 = 4

=> a = 4/2 + 2

a = 2, b = 5

again multiply by 1 and by 2

-4c + 5d = 47

4c - 2d = 26

3d = 21

==> d = 21/3 = 7

and

2c - d = -13

=> 2c -7 = -13

=> 2c = -13 + 7 = -6

=> c = -6/2 = -3

C = -3, D = 7

$Now \: matrix \: A = \binom{2 \: \: \: \: 5}{ - 3 \: \: \: \: 7}$