Name: Properties of Determinants (Part 3)
Uploaded: 2019-07-19
Duration: 3 min 53 s
Description: Suppose we multiply a row of a determinant with a particular value, then what would be the change in the value of the determinant? Watch this video to learn...

Question

Prove that

$|\begin{matrix} a^{2} + 1 & a b & a c \\ a b & b^{2} + 1 & b c \\ c a & c b & c^{2} + 1 \end{matrix}| = 1 + a^{2} + b^{2} + c^{2}$

Accepted Answer

Let

L.H.S= $∆ = |\begin{matrix} a^{2} + 1 & a b & a c \\ a b & b^{2} + 1 & b c \\ a c & b^{2} & c^{2} + 1 \end{matrix}|$

Applying $\frac{R_{1}}{a}, \frac{R_{2}}{b}, \frac{R_{3}}{c}$ we get,

$= |\begin{matrix} a + \frac{1}{a} & b & c \\ a & b + \frac{1}{b} & c \\ a & b & c + \frac{1}{c} \end{matrix}|$

Applying $a C_{1}, b C_{2}, C C_{3}$ we get,

$= |\begin{matrix} a^{2} + 1 & b^{2} & c^{2} \\ a^{2} & b^{2} + 1 & c^{2} \\ a^{2} & b^{2} & c^{2} + 1 \end{matrix}|$

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$ we get,

$= |\begin{matrix} 1 + a^{2} + b^{2} + c^{2} & b^{2} & c^{2} \\ 1 + a^{2} + b^{2} + c^{2} & b^{2} + 1 & c^{2} \\ 1 + a^{2} + b^{2} + c^{2} & b^{2} & c^{2} + 1 \end{matrix}|$

Taking $(1 + a^{2} + b^{2} + c^{2})$ common

$= (1 + a^{2} + b^{2} + c^{2}) |\begin{matrix} 1 & b^{2} & c^{2} \\ 1 & b^{2} + 1 & c^{2} \\ 1 & b^{2} & c^{2} + 1 \end{matrix}|$

Applying $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{1}$ we get

$∆ = (1 + a^{2} + b^{2} + c^{2}) |\begin{matrix} 1 & b^{2} & c^{2} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|$

$= (1 + a^{2} + b^{2} + c^{2}) |\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}|$

$= (1 + a^{2} + b^{2} + c^{2})$

=R.H.S

Hence proved.

Prove that

$|\begin{matrix} a^{2} + 1 & a b & a c \\ a b & b^{2} + 1 & b c \\ c a & c b & c^{2} + 1 \end{matrix}| = 1 + a^{2} + b^{2} + c^{2}$

Important Questions on Determinants

Learn and Prepare for any exam you want !

Prove that a2+1abacabb2+1bccacbc2+1=1+a2+b2+c2

Important Questions on Determinants

Learn and Prepare for any exam you want !

Prove that

$|\begin{matrix} a^{2} + 1 & a b & a c \\ a b & b^{2} + 1 & b c \\ c a & c b & c^{2} + 1 \end{matrix}| = 1 + a^{2} + b^{2} + c^{2}$