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Mathematics

If A=[2a−35],B=[−237b],C=[c9−1−11]A = \begin{bmatrix}[r] 2 & a \ -3 & 5 \end{bmatrix}, B = \begin{bmatrix}[r] -2 & 3 \ 7 & b \end{bmatrix}, C = \begin{bmatrix}[r] c & 9 \ -1 & -11 \end{bmatrix} and 5A + 2B = C, find the values of a, b and c.

Matrices

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Answer

Given, 5A + 2B = C

⇒5[2a−35]+2[−237b]=[c9−1−11]⇒[105a−1525]+[−46142b]=[c9−1−11]⇒[10+(−4)5a+6−15+1425+2b]=[c9−1−11]⇒[65a+6−125+2b]=[c9−1−11]⇒6=c,5a+6=9 and 25+2b=−11⇒c=6,5a=3 and 2b=−11−25⇒c=6,a=35 and 2b=−36⇒c=6,a=35 and b=−18∴a=35,b=−18 and c=6.\Rightarrow 5\begin{bmatrix}[r] 2 & a \ -3 & 5 \end{bmatrix} + 2\begin{bmatrix}[r] -2 & 3 \ 7 & b \end{bmatrix} = \begin{bmatrix}[r] c & 9 \ -1 & -11 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 10 & 5a \ -15 & 25 \end{bmatrix} + \begin{bmatrix}[r] -4 & 6 \ 14 & 2b \end{bmatrix} = \begin{bmatrix}[r] c & 9 \ -1 & -11 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 10 + (-4) & 5a + 6 \ -15 + 14 & 25 + 2b \end{bmatrix} = \begin{bmatrix}[r] c & 9 \ -1 & -11 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 6 & 5a + 6 \ -1 & 25 + 2b \end{bmatrix} = \begin{bmatrix}[r] c & 9 \ -1 & -11 \end{bmatrix} \\[1em] \Rightarrow 6 = c, 5a + 6 = 9 \text{ and } 25 + 2b = -11 \\[0.5em] \Rightarrow c = 6, 5a = 3 \text{ and } 2b = -11 - 25 \\[0.5em] \Rightarrow c = 6, a = \dfrac{3}{5} \text{ and } 2b = -36 \\[0.5em] \Rightarrow c = 6, a = \dfrac{3}{5} \text{ and } b = -18 \\[0.5em] \therefore a = \dfrac{3}{5}, b = -18 \text{ and } c = 6.

Hence, the values are a = 35,\bold{\dfrac{3}{5}}, b = -18 and c = 6.

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