Question

If $\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]$ A = $\left[\begin{array}{ccc}-1& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right]$ Find A

Answer 1

We have XA = 3$\times$3 matrix where X is 3$\times$1. Now XA is conformable for multiplication if the number of columns in pre - factor is equal to the number of rows in the post factor . Keeping in view that the product is 3$\times$3 matrix , we must have matrix A of 1$\times$3 type i.e., A has one row and 3 columns i.e., A is row matrix
Let A = [p.q.r] say of 1$\times$3 type
Now $\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]$ [p q r] = $\left[\begin{array}{ccc}4p& 4q& 4r\\ p& q& r\\ 3p& 3q& 3r\end{array}\right]=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right]$
Clearly p = -1, q = 2 and r = 1 satisfy above
Hence matrix A = (-1 2 1)