and then find matrix C if A + 2B + C = 0.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Question

$A = |\begin{array}{cc} 1 & 1 \\ 2 & 2 \\ 3 & - 1 \end{array}|$ and $B = |\begin{array}{cc} 2 & 1 \\ 1 & 2 \\ - 1 & 0 \end{array}|,$ then find matrix C if A + 2B + C = 0.

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$2 B = 2 |\begin{array}{cc} 2 & 1 \\ 1 & 2 \\ - 1 & 0 \end{array}| = |\begin{array}{cc} 4 & 2 \\ 2 & 4 \\ - 2 & 0 \end{array}|$

$A + 2 B = |\begin{array}{cc} 1 & 1 \\ 2 & 2 \\ 3 & - 1 \end{array}| + |\begin{array}{cc} 4 & 2 \\ 2 & 4 \\ - 2 & 0 \end{array}| = |\begin{array}{cc} 1 + 4 & 2 + 1 \\ 2 + 2 & 4 + 2 \\ ∣ 3 + (- 2) & (- 1) + 0 \end{array}| = |\begin{array}{cc} 5 & 3 \\ 4 & 6 \\ 1 & - 1 \end{array}|$

$C = - (A + 2 B) = |\begin{array}{cc} 5 & 3 \\ 4 & 6 \\ 1 & - 1 \end{array}| \Rightarrow |\begin{array}{cc} - 5 & - 3 \\ - 4 & - 6 \\ - 1 & 1 \end{array}|$

∴ A + 2B + C = 0