[[-22],[27]]=[[2,4],[-3,-7]]Y+[[-8],[7]]

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Given Matrix Equation: We are given the matrix equation [[-22],[27]] = [[2,4],[-3,-7]]Y + [[-8],[7]], and we need to solve for the matrix Y. Let's denote the matrix [[2,4],[-3,-7]] as A and the matrix Y as Y. The equation can be rewritten as: A * Y = [[-22],[27]] - [[-8],[7]]Subtracting Matrices: First, we need to subtract the matrix [[-8],[7]] from the matrix [[-22],[27]] to isolate A * Y on one side of the equation. [[-22],[27]] - [[-8],[7]] = [[-22 - (-8)],[27 - 7]] = [[-22 + 8],[20]] = [[-14],[20]]Finding Inverse of Matrix: Now we have the equation A * Y = [[-14],[20]]. We need to find the inverse of matrix A, which is [[2,4],[-3,-7]], in order to solve for Y.Calculating Determinant: The inverse of a 2x2 matrix [[a,b],[c,d]] is given by (1/(ad-bc)) * [[d,-b],[-c,a]]. Let's calculate the determinant (ad-bc) for matrix A. Determinant = (2 * -7) - (4 * -3) = -14 + 12 = -2Calculating Inverse: Since the determinant is not zero, matrix A is invertible. Now we can calculate the inverse of A. A^(-1) = (1/Determinant) * [[-7,-4],[3,2]] = (-1/2) * [[-7,-4],[3,2]] = [[7/2,4/2],[-3/2,-2/2]] = [[3.5,2],[-1.5,-1]]Multiplying Matrices: Now that we have A^(-1), we can multiply it by the matrix [[-14],[20]] to solve for Y. Y = A^(-1) * [[-14],[20]] = [[3.5,2],[-1.5,-1]] * [[-14],[20]]Matrix Multiplication: Let's perform the matrix multiplication to find Y. Y = [[3.5 * -14 + 2 * 20],[-1.5 * -14 - 1 * 20]] = [[-49 + 40],[-21 - 20]] = [[-9],[-41]]

$\left[\begin{array}{c}-22 \\ 27\end{array}\right]=\left[\begin{array}{cc}2 & 4 \\ -3 & -7\end{array}\right] Y+\left[\begin{array}{c}-8 \\ 7\end{array}\right]$

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[−2227]=[2amp;4−3amp;−7]Y+[−87] \left[\begin{array}{c}-22 \\ 27\end{array}\right]=\left[\begin{array}{cc}2 &amp; 4 \\ -3 &amp; -7\end{array}\right] Y+\left[\begin{array}{c}-8 \\ 7\end{array}\right] [−2227​]=[2−3​amp;4amp;−7​]Y+[−87​]

$\left[\begin{array}{c}-22 \\ 27\end{array}\right]=\left[\begin{array}{cc}2 & 4 \\ -3 & -7\end{array}\right] Y+\left[\begin{array}{c}-8 \\ 7\end{array}\right]$