Question

solve determinant without expanding
4 8 16
2 3 4
5 10 20​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

Evaluate

$\rm \:\:\:\:\begin{gathered}\sf \left | \begin{array}{ccc}4&8&16\\2&3&4\\5&10&20\end{array}\right | \end{gathered}$

Property Used :-

If any two rows or columns of determinant are identical, then determinant value is 0.

Solution :-

• Given determinant is

$\rm \:\:\:\:\begin{gathered}\sf \left | \begin{array}{ccc}4&8&16\\2&3&4\\5&10&20\end{array}\right | \end{gathered}$

• Take out 4 common from Row 1, we have

$\rm \:\: = \:4\:\begin{gathered}\sf \left | \begin{array}{ccc}1&2&4\\2&3&4\\5&10&20\end{array}\right | \end{gathered}$

Again,

• Take out 5 common from Row 3, we have

$\rm \:\: = \:4 \times 5\:\begin{gathered}\sf \left | \begin{array}{ccc}1&2&4\\2&3&4\\1&2&4\end{array}\right | \end{gathered}$

Now,

• Row 1 and Row 3 are identical

and

we know,

• If any 2 rows are identical, the determinant value is 0

So,

$\rm \: \: = \: \: 4 \times 5 \times 0$

$\rm \: \: = \: \: 0$

Hence,

$\red{\bf \:\:\:\:\begin{gathered}\bf \left | \begin{array}{ccc}4&8&16\\2&3&4\\5&10&20\end{array}\right | \end{gathered} = 0}$

Additional Information :-

1. If A is a square matrix of order n, then

$\sf \: |A| = | {A}^{t} |$

$\sf \: | {A}^{ - 1} | = \dfrac{1}{ |A| }$

$\sf \: |kA| = {k}^{n} |A|$

$\sf \: |adjA| = { |A| }^{n - 1}$

2. If any two successive rows or columns are interchanged, the determinant value is multiplied by (-1).

3. If any row or column is zero, then the determinant value is 0.

4. If any row or column is multiplied by any constant 'k', the determinant value is also multiplied by 'k'.

5. The determinant value remains unchanged, if rows or columns are added or subtracted.