Question

if A = [ 2 -1 -4 5 ] and B= [ -3 2 ] , find matrix C such that AC=B

Answer 1

The matrix C = $\left[\begin{array}{ccc}\frac{-13}{6} \\\frac{-4}{3} \end{array}\right]$ when A = $\left[\begin{array}{ccc}2&-1\\-4&5\end{array}\right]$ and B = $\left[\begin{array}{ccc}-3\\2\end{array}\right]$ such that AC = B and by multiplication matrix with variables.

Given that,

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

We have to find the matrix C such that AC = B

We know that,

AC = $\left[\begin{array}{ccc}2&-1\\-4&5\end{array}\right]$$\left[\begin{array}{ccc}-3\\2\end{array}\right]$  (by multiplication matrix)

We get

AC = $\left[\begin{array}{ccc}2x-y\\-4x+5y\end{array}\right]$

From the given

AC = B

$\left[\begin{array}{ccc}2x-y\\-4x+5y\end{array}\right]$ = $\left[\begin{array}{ccc}-3\\2\end{array}\right]$

2x-y = -3  ------->equation(1)

and

-4x+5y = 2  ------->equation(2)

Solve equation(1) and equation(2)

y = 2x+3

Substitute y in equation(2)

-4x+5(2x+3) = 2

-4x + 10x + 15 = 2

6x = 2 - 15

6x = -13

x = $\frac{-13}{6}$

Substitute x value in y

y = 2$(\frac{-13}{6})$ +3

y = $\frac{-4}{3}$

Therefore, The matrix C = $\left[\begin{array}{ccc}\frac{-13}{6} \\\frac{-4}{3} \end{array}\right]$

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