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Mathematics Question on Matrices

Find the matrix X so that X[123 456]=[−7−8−9 246]X\begin{bmatrix}1&2&3\\\ 4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\\ 2&4&6\end{bmatrix}

Answer

The correct answer is [1−2\20]\begin{bmatrix}1&-2\\\2&0\end{bmatrix}
It is given that X[123 456]=[−7−8−9 246]X\begin{bmatrix}1&2&3\\\ 4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\\ 2&4&6\end{bmatrix}
The matrix given on the R.H.S. of the equation is a 2×32 \times 3 matrix and the one given on the L.H.S. of the equation is a 2×32 \times 3 matrix. Therefore, X has to be a 2×22 \times 2 matrix.
Now, let X=[ac\bd]X=\begin{bmatrix}a&c\\\b&d\end{bmatrix}
Therefore, we have:
[ac\bd][123 456]=[−7−8−9 246]\begin{bmatrix}a&c\\\b&d\end{bmatrix}\begin{bmatrix}1&2&3\\\ 4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\\ 2&4&6\end{bmatrix}
  ⟹  [a+4c2a+5c3a+6c b+4d2b+5d3b+6d]=[−7−8−9 246]\implies\begin{bmatrix}a+4c& 2a+5c& 3a+6c\\\ b+4d& 2b+5d& 3b+6d\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\\ 2&4&6\end{bmatrix}
Equating the corresponding elements of the two matrices, we have:
a+4c=−7,2a+5c=−8,3a+6c=−9a+4c=-7,2a+5c=-8,3a+6c=-9
b+4d=2,2b+5d=4,3b+6d=6b+4d=2,2b+5d =4,3b+6d=6
Now a+4c=−7a+4c=-7
=  ⟹  a=−7−4c=\implies a=-7-4c
2a+5c=−8  ⟹  2(−7−4c)+5c=−82a+5c=-8\implies 2(-7-4c)+5c=-8
  ⟹  −14−8c+5c=−8\implies-14-8c+5c=-8
  ⟹  c=−2\implies c=-2
so a=−7−4c  ⟹  a=−7−4(−2)=−7+8=1a=-7-4c\implies a=-7-4(-2)=-7+8=1
Now b+4d=2b+4d=2
=  ⟹  b=2−4d=\implies b=2-4d
Now 2b+5d=4  ⟹  4−8d+5d=42b+5d =4\implies 4-8d+5d=4
  ⟹  −3d=0\implies-3d=0
  ⟹  d=0\implies d=0
so b=2−4(0)=2b=2-4(0)=2
Thus, a=1,b=2,c=−2,d=0a = 1, b = 2, c = −2, d = 0
Hence, the required matrix X is [1−2\20]\begin{bmatrix}1&-2\\\2&0\end{bmatrix}