Find the matrix A such that

$A=\left[\begin{array}{lll}a & b & c \\ d & e & f\end{array}\right]$

$\left[\begin{array}{cc}2 & -1 \\ 1 & 0 \\ -3 & 4\end{array}\right]\left[\begin{array}{lll}a & b & c \\ d & e & f\end{array}\right]=\left[\begin{array}{ccc}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]$

$\left[\begin{array}{ccc}2 a-d & 2 b-e & 2 c-f \\ a & b & c \\ -3 a+4 d & -3 b+4 e & -3 c+4 f\end{array}\right]=\left[\begin{array}{ccc}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]$

Let,

Now, by equality of matrices, we get

a = 1, b = -2, c = -5

And,

2a – d = -1 ⇒ d = 2a + 1 = 3;

2b – e = -8 ⇒ e = 2(-2) + 8 = 4

2c – f = -10 ⇒ f = 2c + 10 = 0

Thus,

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