Question

Prove that
| a a+b a+b+c |
|3a 4a+3b 5a+4b+3c | =-a³
|6a 9a+6b 11a+9b+6c |

Answer 1

Answer:

$\bf\:\left|\begin{array}{ccc}a&a+b&a+b+c\\3a&4a+3b&5a+4b+3c\\6a&9a+6b&11a+9b+6c\end{array}\right|=-a^3$

Step-by-step explanation:

Prove that

| a a+b a+b+c |

|3a 4a+3b 5a+4b+3c | =-a³

|6a 9a+6b 11a+9b+6c |

$\left|\begin{array}{ccc}a&a+b&a+b+c\\3a&4a+3b&5a+4b+3c\\6a&9a+6b&11a+9b+6c\end{array}\right|$

$=\left|\begin{array}{ccc}a&a+b&a+b+c\\3a&4a+3b&5a+4b+3c\\0&a&a+b\end{array}\right|$ $R_3\implies\:R_3-2R_2$

$=\left|\begin{array}{ccc}a&a+b&a+b+c\\0&a&2a+b\\0&a&a+b\end{array}\right|$ $R_2\implies\:R_2-3R_1$

$=\left|\begin{array}{ccc}a&a+b&a+b+c\\0&0&a\\0&a&a+b\end{array}\right|$ $R_2\implies\:R_2-R_3$

Expanding along Column1

$=a(0-a^2)-0+0$

$=a(-a^2)$

$=-a^3$

$\bf\:\implies\left|\begin{array}{ccc}a&a+b&a+b+c\\3a&4a+3b&5a+4b+3c\\6a&9a+6b&11a+9b+6c\end{array}\right|=-a^3$