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Find the matrix A such that$\begin{bmatrix}2 & -1\\1 & 0\\-3 & 4\end{bmatrix}\;A=\begin{bmatrix}-1 & -8 & -10\\1& -2 & -5\\9 & 22 &15\end{bmatrix}.$

Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
• If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij}=B_{ij}$, then any element $a_{ij}$ in matrix A is equal to corresponding element $b_{ij}$ in matrix B.
• We can then match the corresponding elements and solve the resulting equations to find the values of the unknown variables.
Step1:
Given:
Let $B=\begin{bmatrix}2 & -1\\1 & 0\\-3 & 4\end{bmatrix}\;C=\begin{bmatrix}-1 & -8 & -10\\1& -2 & -5\\9 & 22 &15\end{bmatrix}.$

$\Rightarrow BA=C.$

Since B is a $3\times 2$ matrix.

$\Rightarrow$ A should be $2\times 3$ matrix.

$\Rightarrow A =\begin{bmatrix}a & b & c\\d & e & f\end{bmatrix}$
Step2:
$BA=\begin{bmatrix}2 & -1\\1 &0\\-3 & 4\end{bmatrix}\begin{bmatrix}a & b & c\\d & e & f\end{bmatrix}$

$\Rightarrow \begin{bmatrix}2a-d & 2b-e & 2c-f\\a+0 & b+0 & c+0\\-3a+4d& -3b+4c & -3c+4f\end{bmatrix}$

$\Rightarrow \begin{bmatrix}2a-d & 2b-e & 2c-f\\a & b & c\\-3a+4d& -3b+4c & -3c+4f\end{bmatrix}=\begin{bmatrix}-1 & -8 & -10\\1 & -2 & -5\\9 & 22 & 15\end{bmatrix}$

$\Rightarrow$ since the given two matrices are equal .Thier corresponding elements should be equal.
Step3:
$\Rightarrow$ From the 2$^nd$ row of the two matrices we have,
a=1,b=-2,c=-5.
By comparing the first row we have
2a-d=-1----(1)
2b-e=-8-----(2)
2c-f=-10------(3)
Step4:
From equation (1) we have
2a-d=-1.
Substitute the value of a in equation (1)
2(1)-d=-1
2-d=-1
-d=-1-2.
-d=-3.
d=3
Step5:
From equation (2) we have
2b-e=-8
Substitute the value of b in equation (2)
2(-2)-e=-8
-4-e=-8
-e=-8+4.
-e=-4
e=4
Step6:
From equation (3) we have
2c-f=-10
Substitute the value of c in equation (3)
2(-5)-f=-10
-10-f=-10
-f=-10+10
f=0
Hence the matrix A is
$\begin{bmatrix}a & b & c\\d & e & f\end{bmatrix}=\begin{bmatrix}1 & -2 & -5\\3 & 4 & 0\end{bmatrix}$

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