Solve the matrix equations-i - x 1- 1 0-2-3 x 5-0ii -1 21- 12020110202 x -0iii - x -5-1- 102021203 - 41-0

I #160; #160;x110 2 3x5=0 #8658;x 20 3x5=0 #8658;x 2 3x5=0 #8658;x2 2x 15=0 #8658;x2 2x 15=0 #8658;x2 5x+3x 15=0 #8658;xx 5+3x 5=0 #8658;x 5x+ ...

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Byju's Answer

Standard XII

Mathematics

Manipulation of a linear equation

Solve the mat...

Question

Solve the matrix equations:
(i) $[x 1] [\begin{matrix} 1 & 0 \\ - 2 & - 3 \end{matrix}] [\begin{matrix} x \\ 5 \end{matrix}] = 0$

(ii) $[1 2 1] [\begin{matrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{matrix}] [\begin{matrix} 0 \\ 2 \\ x \end{matrix}] = 0$

(iii) $[x - 5 - 1] [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}] [\begin{matrix} x \\ 4 \\ 1 \end{matrix}] = 0$

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Solution

$(i) [\begin{matrix} x & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 2 & - 3 \end{matrix}] [\begin{matrix} x \\ 5 \end{matrix}] = 0 \Rightarrow [\begin{matrix} x - 2 & 0 - 3 \end{matrix}] [\begin{matrix} x \\ 5 \end{matrix}] = 0 \Rightarrow [\begin{matrix} x - 2 & - 3 \end{matrix}] [\begin{matrix} x \\ 5 \end{matrix}] = 0 \Rightarrow [\begin{matrix} x^{2} - 2 x - 15 \end{matrix}] = 0 \Rightarrow x^{2} - 2 x - 15 = 0 \Rightarrow x^{2} - 5 x + 3 x - 15 = 0 \Rightarrow x (x - 5) + 3 (x - 5) = 0 \Rightarrow (x - 5) (x + 3) = 0 \Rightarrow x - 5 = 0 or x + 3 = 0 \Rightarrow x = 5 or x = - 3$

$(i i) [\begin{matrix} 1 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{matrix}] [\begin{matrix} 0 \\ 2 \\ x \end{matrix}] = 0 \Rightarrow [\begin{matrix} 1 + 4 + 1 & 2 + 0 + 0 & 0 + 2 + 2 \end{matrix}] [\begin{matrix} 0 \\ 2 \\ x \end{matrix}] = 0 \Rightarrow [\begin{matrix} 6 & 2 & 4 \end{matrix}] [\begin{matrix} 0 \\ 2 \\ x \end{matrix}] = 0 \Rightarrow [\begin{matrix} 0 + 4 + 4 x \end{matrix}] = 0 \Rightarrow 4 + 4 x = 0 \Rightarrow 4 x = - 4 ∴ x = \frac{- 4}{4} = - 1$

$(i i i) [\begin{matrix} x & - 5 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}] [\begin{matrix} x \\ 4 \\ 1 \end{matrix}] = 0 \Rightarrow [\begin{matrix} x - 0 - 2 & 0 - 10 - 0 & 2 x - 5 - 3 \end{matrix}] [\begin{matrix} x \\ 4 \\ 1 \end{matrix}] = 0 \Rightarrow [\begin{matrix} x - 2 & - 10 & 2 x - 8 \end{matrix}] [\begin{matrix} x \\ 4 \\ 1 \end{matrix}] = 0 \Rightarrow [\begin{matrix} x^{2} - 2 x - 40 + 2 x - 8 \end{matrix}] = 0 \Rightarrow x^{2} - 48 = 0 \Rightarrow x^{2} = 48 \Rightarrow x = \pm \sqrt{48}$

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Solve the matrix equations: (i) [x 1] 1 0-2-3x5=0 (ii) [1 2 1]12020110202x=0 (iii) [x-5-1]102021203x41=0