Matrices & Determinants
Grade 12
Question:
If $A = \begin{bmatrix} 1 & 3 \\ 3 & 2 \\ 2 & 5 \end{bmatrix}$ & $B = \begin{bmatrix} -1 & -2 \\ 0 & 5 \\ 3 & 1 \end{bmatrix}$ and $A + B - D = O$ (zero matrix), then $D$ matrix will be-
$\begin{bmatrix} 0 & 2 \\ 3 & 7 \\ 6 & 5 \end{bmatrix}$
$\begin{bmatrix} 0 & 2 \\ 3 & 7 \\ 5 & 6 \end{bmatrix}$
$\begin{bmatrix} 0 & 1 \\ 3 & 7 \\ 5 & 6 \end{bmatrix}$
$\begin{bmatrix} 0 & -2 \\ -3 & -7 \\ -5 & -6 \end{bmatrix}$
Step-by-Step Solution
Key Concept: Matrix addition and subtraction are performed element-wise on corresponding entries, provided all matrices in the equation have the same dimensions.
Let $D = \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}$<br>$\therefore A + B - D = \begin{bmatrix} 1 & 3 \\ 3 & 2 \\ 2 & 5 \end{bmatrix} + \begin{bmatrix} -1 & -2 \\ 0 & 5 \\ 3 & 1 \end{bmatrix} - \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix} \Rightarrow \begin{bmatrix} 1-1-a & 3-2-b \\ 3+0-c & 2+5-d \\ 2+3-e & 5+1-f \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}$<br>$\Rightarrow -a = 0 \Rightarrow a = 0, 1-b = 0 \Rightarrow b = 1,$<br>$\Rightarrow 3-c = 0 \Rightarrow c = 3, 7-d = 0 \Rightarrow d = 7,$<br>$\Rightarrow 5-e = 0 \Rightarrow e = 5, 6-f = 0 \Rightarrow f = 6$<br>$\therefore D = \begin{bmatrix} 0 & 1 \\ 3 & 7 \\ 5 & 6 \end{bmatrix}$
Correct Answer: C