Q45. Find a third order determinant i... [FREE SOLUTION]

Chapter 4: Q45. (page 187)

Find a third order determinant in which no element is 0, but for which the determinant is 0.

Short Answer

Expert verified

Thus, a third order determinant in which no element is 0, but for which the determinant is 0 is $|\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}|$

Step by step solution

- Write a third order determinant

A third order determinant is determinant of a square matrix of order 3.

It can be simplified as shown below

$|\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}| = a |\begin{matrix} e & f \\ h & i \end{matrix}| 鈭� b |\begin{matrix} d & f \\ g & i \end{matrix}| + c |\begin{matrix} d & e \\ g & h \end{matrix}|$

- Write a third order determinant in which no element is 0

Substitute 1 for all elements in a third order determinant

$|\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}|$

- Simplify 111111111

To evaluate this determinant expand by using minors

$\begin{matrix} |\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}| = 1 |\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}| 鈭� 1 |\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}| + 1 |\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}| \\ = 1 (1 鈭� 1) 鈭� 1 (1 鈭� 1) + 1 (1 鈭� 1) \\ = 0 鈭� 0 + 0 \\ = 0 \end{matrix}$