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Question: If \(A = \left[ {\begin{array}{*{20}{c}} 1&0&1 \\\ 0&1&2 \\\ 0&0&4 \end{array}} \right]\)...

If A = \left[ {\begin{array}{*{20}{c}} 1&0&1 \\\ 0&1&2 \\\ 0&0&4 \end{array}} \right] then show that∣3A∣=27∣A∣\left| {3A} \right| = 27\left| A \right|.

Explanation

Solution

Hint- ∣3A∣\left| {3A} \right|means first A matrix is multiplied with 3 and then it’s determinant is to be found. Evaluate each LHS and RHS separately, to prove.
We have given that A = \left[ {\begin{array}{*{20}{c}} 1&0&1 \\\ 0&1&2 \\\ 0&0&4 \end{array}} \right]
Now we show that ∣3A∣=27∣A∣\left| {3A} \right| = 27\left| A \right|
First let’s calculate the LHS part so 3A = 3\left[ {\begin{array}{*{20}{c}} 1&0&1 \\\ 0&1&2 \\\ 0&0&4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3&0&3 \\\ 0&3&6 \\\ 0&0&{12} \end{array}} \right]
Now the determinant of 3A that is ∣3A∣\left| {3A} \right|
\left| {\begin{array}{*{20}{c}} 3&0&3 \\\ 0&3&6 \\\ 0&0&{12} \end{array}} \right|=[3(3×12−0×6)−0(0×12−0×6)+3(0×0−0×3)]\left[ {3\left( {3 \times 12 - 0 \times 6} \right) - 0\left( {0 \times 12 - 0 \times 6} \right) + 3\left( {0 \times 0 - 0 \times 3} \right)} \right]
On simplifying we get
\left| {\begin{array}{*{20}{c}} 3&0&3 \\\ 0&3&6 \\\ 0&0&{12} \end{array}} \right| = 3 \times 36 = 108………………………………….. (1)
Now we have to find 27∣A∣27\left| A \right|
That is 27\left| {\begin{array}{*{20}{c}} 1&0&1 \\\ 0&1&2 \\\ 0&0&4 \end{array}} \right| = 27\left[ {1 \times \left( {1 \times 4 - 0 \times 2} \right) - 0\left( {0 \times 4 - 0 \times 2} \right) + 1\left( {0 \times 0 - 0 \times 1} \right)} \right]
On simplifying we get

27\left| {\begin{array}{*{20}{c}} 1&0&1 \\\ 0&1&2 \\\ 0&0&4 \end{array}} \right| = 27 \times 4 = 108……………………………… (2)
Clearly equation (1) is equal to equation (2) thus we can say that ∣3A∣=27∣A∣\left| {3A} \right| = 27\left| A \right|
Hence proved.
Note- The key concept involved here is that we need to understand the basics of determinant evaluation: the quantity inside the determinant resembles a matrix , if it is multiplied with a scalar then the determinant of that scalar multiplied matrix is to be found. However if a scalar is multiplied with a determinant then simply the product of determinant and scalar number is to be evaluated.