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Mathematics Question on types of matrices

If A=[13 32 25]A = \begin{bmatrix}1&3\\\ 3&2\\\ 2&5\end{bmatrix} and B=[−1−2 05 31]B = \begin{bmatrix}-1&-2\\\ 0&5\\\ 3&1\end{bmatrix} and A+B−D=0A + B - D = 0 (zero matrix), then DD matrix will be -

A

[02 37 65] \begin{bmatrix} 0 & 2 \\\ 3&7 \\\ 6 &5\end{bmatrix}

B

[02 37 56] \begin{bmatrix} 0 & 2 \\\ 3&7 \\\ 5 & 6 \end{bmatrix}

C

[01 37 56] \begin{bmatrix} 0 &1 \\\ 3&7 \\\ 5 & 6 \end{bmatrix}

D

[0−2 −3−7 −5−6] \begin{bmatrix} 0 & -2 \\\ -3 & -7 \\\ -5 & -6 \end{bmatrix}

Answer

[01 37 56] \begin{bmatrix} 0 &1 \\\ 3&7 \\\ 5 & 6 \end{bmatrix}

Explanation

Solution

Let D=[ab cd ef]D = \begin{bmatrix}a&b\\\ c&d\\\ e&f\end{bmatrix}
∴A+B−C=[13 32 25]+[−1−2 05 31]−[ab cd ef]\therefore A + B - C = \begin{bmatrix}1&3\\\ 3&2\\\ 2&5\end{bmatrix}+ \begin{bmatrix}-1&-2\\\ 0&5\\\ 3&1\end{bmatrix} - \begin{bmatrix}a&b\\\ c&d\\\ e&f\end{bmatrix}
⇒[1−1−a3−2−b 3+0−c2+5−d 2+3−e5+1−f]=[00 00 00]\Rightarrow \begin{bmatrix}1-1-a&3-2-b\\\ 3+0-c&2+5-d\\\ 2+3-e&5+1-f\end{bmatrix} =\begin{bmatrix}0&0\\\ 0&0\\\ 0&0\end{bmatrix}
−a=0⇒a=0,1−b=0-a=0 \Rightarrow a=0,1-b=0
⇒b=1\Rightarrow b=1
3−c=0⇒c=3,7−d=03-c=0 \Rightarrow c=3,7-d=0
⇒d=7\Rightarrow d=7,
5−e=0⇒e=5,6−f=05-e=0 \Rightarrow e=5,6-f=0
⇒f=6\Rightarrow f=6,
∴D=[01 37 56]\therefore D = \begin{bmatrix}0&1\\\ 3&7\\\ 5&6\end{bmatrix}