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If [{:(,4,-2),(,4,0):}]+3A=[{:(,-2,-2),(...

If `[{:(,4,-2),(,4,0):}]+3A=[{:(,-2,-2),(,1,-3):}]` find A.

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To solve the equation \([ \begin{pmatrix} 4 & -2 \\ 4 & 0 \end{pmatrix} ] + 3A = [ \begin{pmatrix} -2 & -2 \\ 1 & -3 \end{pmatrix} ]\), we will follow these steps: ### Step 1: Isolate \(3A\) We start by isolating \(3A\) on one side of the equation. We can do this by subtracting the matrix \([ \begin{pmatrix} 4 & -2 \\ 4 & 0 \end{pmatrix} ]\) from both sides: \[ 3A = [ \begin{pmatrix} -2 & -2 \\ 1 & -3 \end{pmatrix} ] - [ \begin{pmatrix} 4 & -2 \\ 4 & 0 \end{pmatrix} ] \] ### Step 2: Perform the subtraction Now we will perform the matrix subtraction element-wise: \[ 3A = [ \begin{pmatrix} -2 - 4 & -2 - (-2) \\ 1 - 4 & -3 - 0 \end{pmatrix} ] \] Calculating each element: - First element: \(-2 - 4 = -6\) - Second element: \(-2 + 2 = 0\) - Third element: \(1 - 4 = -3\) - Fourth element: \(-3 - 0 = -3\) So we have: \[ 3A = [ \begin{pmatrix} -6 & 0 \\ -3 & -3 \end{pmatrix} ] \] ### Step 3: Solve for \(A\) To find \(A\), we need to divide each element of the matrix \(3A\) by 3: \[ A = \frac{1}{3} [ \begin{pmatrix} -6 & 0 \\ -3 & -3 \end{pmatrix} ] \] Calculating each element: - First element: \(-6 / 3 = -2\) - Second element: \(0 / 3 = 0\) - Third element: \(-3 / 3 = -1\) - Fourth element: \(-3 / 3 = -1\) Thus, we find: \[ A = [ \begin{pmatrix} -2 & 0 \\ -1 & -1 \end{pmatrix} ] \] ### Final Answer The matrix \(A\) is: \[ A = [ \begin{pmatrix} -2 & 0 \\ -1 & -1 \end{pmatrix} ] \]
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