If and , find a matrix C such that 3A + 5B + 2C is a null matrix.

Engineering

Mathematics

Algebra of Matrices

Question

If $A = [\begin{array}{cc} 1 & 5 \\ 7 & 12 \end{array}]$ and $B = [\begin{array}{cc} 9 & 1 \\ 7 & 8 \end{array}]$ , find a matrix C such that 3A + 5B + 2C is a null matrix.

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We have, $A = [\begin{array}{cc} 1 & 5 \\ 7 & 12 \end{array}]$ and $B = [\begin{array}{cc} 9 & 1 \\ 7 & 8 \end{array}]$

Let

C = [\begin{array}{cc} a & b \\ c & d \end{array}]

∴ 3A + 5B + 2C = 0

\Rightarrow [\begin{array}{cc} 3 & 15 \\ 21 & 36 \end{array}] + [\begin{array}{cc} 45 & 5 \\ 35 & 40 \end{array}] + [\begin{array}{cc} 2 a & 2 b \\ 2 c & 2 d \end{array}] = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

\Rightarrow [\begin{array}{cc} 48 + 2 a & 20 + 2 b \\ 56 + 2 c & 76 + 2 c \end{array}] = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]

⇒ 2a + 48 = 0 ⇒ a = – 24

20 + 2b = 0 ⇒ b = – 10

56 + 2c = 0 ⇒ c = – 28

and 76 + 2d = 0 ⇒ – d = – 38

C = [\begin{array}{cc} - 24 & - 10 \\ - 28 & - 38 \end{array}]