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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

01

- 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

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

02

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

Substitute 1 for all elements in a third order determinant

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

03

- Simplify 111111111

To evaluate this determinant expand by using minors

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

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Most popular questions from this chapter

For exercises 26 and 27, use the advertisement shown at the right.

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$[\begin{matrix} 7 & 3 \\ 0 & 2 \\ 5 & 5 \end{matrix}] \cdot [\begin{matrix} - 2 & 1 & 4 \\ 3 & - 5 & 2 \\ 4 & 3 & 1 \end{matrix}]$

See all solutions

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