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

Without expanding at any stage, find the value of the determinant: $∆ = |\begin{array}{l} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{array}|$ .

Accepted Answer

Given that, we have: $∆ = |\begin{array}{l} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{array}|$

Operating $C_{3} \to C_{3} + C_{2}$ , then $∆ = |\begin{array}{l} 20 & a & a + b + c \\ 20 & b & a + b + c \\ 20 & c & a + b + c \end{array}|$

$\Rightarrow ∆ = 20 (a + b + c) |\begin{array}{l} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \end{array}|$ [Taking common factor $20$ and $a + b + c$ from $C_{1} and C_{3}$ ]

$\Rightarrow ∆ = 20 (a + b + c) (0)$ [ $∵ C_{1} and C_{3}$ are identical columns]

$\Rightarrow ∆ = 0$

Hence, $∆ = |\begin{array}{l} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{array}| = 0$ .

Without expanding at any stage, find the value of the determinant: $∆ = |\begin{array}{l} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{array}|$ .

Important Questions on Determinants

Learn and Prepare for any exam you want !

Without expanding at any stage, find the value of the determinant: ∆=20ab+c20ba+c20ca+b.

Important Questions on Determinants

Learn and Prepare for any exam you want !

Without expanding at any stage, find the value of the determinant: $∆ = |\begin{array}{l} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{array}|$ .