Question

Use the properties of determinants to evaluate the following determinant:

(b+c)^2 a^2 a^2
b^2 (c+a^)2 b^2
c^2 c^2 (a+b)^2

Answer 1

Answer on Question #42306 – Math - Linear Algebra

Use the properties of determinants to evaluate the following determinant:

$$\left| \begin{array}{c c c} (b + c) ^ {2} & a ^ {2} & a ^ {2} \\ b ^ {2} & (c + a) ^ {2} & b ^ {2} \\ c ^ {2} & c ^ {2} & (a + b) ^ {2} \end{array} \right|$$

Solution

Let

$$\Delta = \left| \begin{array}{c c c} (b + c) ^ {2} & a ^ {2} & a ^ {2} \\ b ^ {2} & (c + a) ^ {2} & b ^ {2} \\ c ^ {2} & c ^ {2} & (a + b) ^ {2} \end{array} \right|.$$

Applying $C_1 \to C_1 - C_3$ and $C_2 \to C_2 - C_3$, we get,

$$\Delta = \left| \begin{array}{c c c} (b + c) ^ {2} - a ^ {2} & 0 & a ^ {2} \\ 0 & (c + a) ^ {2} - b ^ {2} & b ^ {2} \\ c ^ {2} - (a + b) ^ {2} & c ^ {2} - (a + b) ^ {2} & (a + b) ^ {2} \end{array} \right| = (a + b + c) ^ {2} \left| \begin{array}{c c c} b + c - a & 0 & a ^ {2} \\ 0 & c + a - b & b ^ {2} \\ c - a - b & c - a - b & (a + b) ^ {2} \end{array} \right|$$

here, take common $(a + b + c)$ from $C_1$ & $C_2$.

Now, applying $R_3 \to R_3 - (R_1 + R_2)$, we get,

$$\Delta = (a + b + c) ^ {2} \left| \begin{array}{c c c} b + c - a & 0 & a ^ {2} \\ 0 & c + a - b & b ^ {2} \\ - 2 b & - 2 a & 2 a b \end{array} \right| = \frac {(a + b + c) ^ {2}}{a b} \left| \begin{array}{c c c} a b + a c - a ^ {2} & 0 & a ^ {2} \\ 0 & c + a - b & b ^ {2} \\ - 2 a b & - 2 a b & 2 a b \end{array} \right|,$$

applying $C_1 \to aC_1$ & $C_2 \to bC_2$

$$\Delta = \frac {(a + b + c) ^ {2}}{a b} \left| \begin{array}{c c c} a b + a c & a ^ {2} & a ^ {2} \\ b ^ {2} & b c + b a & b ^ {2} \\ 0 & 0 & 2 a b \end{array} \right|,$$

applying $C_1 \to C_1 + C_3$ & $C_2 \to C_2 + C_3$

$$\Delta = \frac {(a + b + c) ^ {2}}{a b} a b \cdot 2 a b \left| \begin{array}{c c c} b + c & a & a \\ b & c + a & b \\ 0 & 0 & 1 \end{array} \right|,$$

here, take $a, b, c$ common from $R_1, R_2$ & $R_3$

$$\Delta = 2 a b (a + b + c) ^ {2} \left| \begin{array}{c c} b + c & a \\ b & c + a \end{array} \right|,$$

expand along $R_3$

$$\begin{array}{l} \Delta = 2 a b (a + b + c) ^ {2} \left[ (b + c) (c + a) - a b \right] = 2 a b (a + b + c) ^ {2} [ a b + a c + b c + c ^ {2} - a b ] \\ = 2 a b c (a + b + c) ^ {2} (a + b + c) = 2 a b c (a + b + c) ^ {3}. \end{array}$$

Answer: $2a b c(a + b + c)^{3}$.

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