Question

If P= [(6,-2), (4,-6)] and Q = [(5,3), (2,0)], find the matrix M such that , 2Q-3P-3M = 0

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:M\:=\:\left[\:\begin{array}{cc}\sf\:-\:\dfrac{8}{3} & \sf\:4\\\sf\:-\:\dfrac{8}{3} & \sf\:6\\\end{array}\:\right]\:}}}$

Step-by-step-explanation:

NOTE: Refer to the attachment for the answer in app.

We have given two matrices P and Q.

We have to find the matrix M for which 2Q - 3P - 3M = 0.

Now,

$\displaystyle{\sf\:P\:=\:\left[\:\begin{array}{cc}\sf\:6 & \sf\:-\:2\\\sf\:4 & \sf\:-\:6\\\end{array}\:\right]}$

We have to find 3P.

$\displaystyle{\therefore\:\sf\:3P\:=\:3\:\times\:\left[\:\begin{array}{cc}\sf\:6 & \sf\:-\:2\\\sf\:4 & \sf\:-\:6\\\end{array}\:\right]}$

$\displaystyle{\implies\sf\:3P\:=\:\left[\:\begin{array}{cc}\sf\:6\:\times\:3 & \sf\:-\:2\:\times\:3\\\sf\:4\:\times\:3 & \sf\:-\:6\:\times\:3\\\end{array}\:\right]}$

$\displaystyle{\implies\:\boxed{\sf\:3P\:=\:\left[\:\begin{array}{cc}\sf\:18 & \sf\:-\:6\\\sf\:12 & \sf\:-\:18\\\end{array}\:\right]}}$

Now,

$\displaystyle{\sf\:Q\:=\:\left[\:\begin{array}{cc}\sf\:5 & \sf\:3\\\sf\:2 & \sf\:0\\\end{array}\:\right]}$

We have to find 2Q.

$\displaystyle{\therefore\:\sf\:2Q\:=\:2\:\times\:\left[\:\begin{array}{cc}\sf\:5 & \sf\:3\\\sf\:2 & \sf\:0\\\end{array}\:\right]}$

$\displaystyle{\implies\sf\:2Q\:=\:\left[\:\begin{array}{cc}\sf\:5\:\times\:2 & \sf\:3\:\times\:2\\\sf\:2\:\times\:2 & \sf\:0\:\times\:3\\\end{array}\:\right]}$

$\displaystyle{\implies\:\boxed{\sf\:2Q\:=\:\left[\:\begin{array}{cc}\sf\:10 & \sf\:6\\\sf\:4 & \sf\:0\\\end{array}\:\right]}}$

Now,

$\displaystyle{\sf\:2Q\:-\:3P\:-\:3M\:=\:0}$

$\displaystyle{\implies\sf\:3M\:=\:2Q\:-\:3P}$

$\displaystyle{\implies\sf\:3M\:=\:\left[\:\begin{array}{cc}\sf\:10 & \sf\:6\\\sf\:4 & \sf\:0\\\end{array}\:\right]\:-\:\left[\:\begin{array}{cc}\sf\:18 & \sf\:-\:6\\\sf\:12 & \sf\:-\:18\\\end{array}\:\right]}$

$\displaystyle{\implies\sf\:3M\:=\:\left[\:\begin{array}{cc}\sf\:10\:-\:18 & \sf\:6\:-\:(\:-\:6\:)\\\sf\:4\:-\:12 & \sf\:0\:-\:(\:-\:18\:)\\\end{array}\:\right]}$

$\displaystyle{\implies\sf\:3M\:=\:\left[\:\begin{array}{cc}\sf\:-\:8 & \sf\:12\\\sf\:-\:8 & \sf\:18\\\end{array}\:\right]}$

$\displaystyle{\implies\sf\:M\:=\:\dfrac{1}{3}\:\times\:\left[\:\begin{array}{cc}\sf\:-\:8 & \sf\:12\\\sf\:-\:8 & \sf\:18\\\end{array}\:\right]}$

$\displaystyle{\implies\sf\:M\:=\:\left[\:\begin{array}{cc}\sf\:-\:\dfrac{8}{3} & \sf\:\dfrac{12}{3}\\\sf\:-\:\dfrac{8}{3} & \sf\:\dfrac{18}{3}\\\end{array}\:\right]}$

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:M\:=\:\left[\:\begin{array}{cc}\sf\:-\:\dfrac{8}{3} & \sf\:4\\\sf\:-\:\dfrac{8}{3} & \sf\:6\\\end{array}\:\right]\:}}}}$

Answer 2

$\small\red{Answer \: refer \: to \: the \: attachment}$

$\small\pink{Thank \: You}$