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Mathematics

Find X and Y if

X + Y=[7025] and X - Y=[3003].\text{X + Y} = \begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix} \text{ and X - Y} = \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix}.

Matrices

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Answer

Given,

X + Y = [7025]\begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix}        (…Eq1)

X - Y = [3003]\begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix}         (…Eq2)

Adding both the equations,

⇒X+Y+X−Y=[7025]+[3003]⇒2X=[7+30+02+05+3]⇒X=12[10028]⇒X=[5014]\Rightarrow X + Y + X - Y = \begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix} + \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix} \\[1em] \Rightarrow 2X = \begin{bmatrix} 7 + 3 & 0 + 0 \ 2 + 0 & 5 + 3 \end{bmatrix} \\[1em] \Rightarrow X = \dfrac{1}{2}\begin{bmatrix} 10 & 0 \ 2 & 8 \end{bmatrix} \\[1em] \Rightarrow X = \begin{bmatrix} 5 & 0 \ 1 & 4 \end{bmatrix} \\[1em]

From Eq 1 we get,

⇒Y=[7025]−X=[7025]−[5014]=[7−50−02−15−4]=[2011]∴X=[5014],Y=[2011].\Rightarrow Y = \begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix} - X \\[1em] = \begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix} - \begin{bmatrix} 5 & 0 \ 1 & 4 \end{bmatrix} \\[1em] = \begin{bmatrix} 7 - 5 & 0 - 0 \ 2 - 1 & 5 - 4 \end{bmatrix} \\[1em] = \begin{bmatrix} 2 & 0 \ 1 & 1 \end{bmatrix} \\[1.5em] \therefore X = \begin{bmatrix} 5 & 0 \ 1 & 4 \end{bmatrix} , Y = \begin{bmatrix} 2 & 0 \ 1 & 1 \end{bmatrix}.

Hence, X=[5014] and Y=[2011].X = \begin{bmatrix} 5 & 0 \ 1 & 4 \end{bmatrix} \text{ and } Y = \begin{bmatrix} 2 & 0 \ 1 & 1 \end{bmatrix}.

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