Question

If A is \[{\text{3}} \times {\text{3}}\] matrix and \[{\text{|3A| = K|A|}}\] then \[{\text{K = }}\]

A) 9

B) 6

C) 1

D) 27

Answer 1

Hint: Here first we need to find the matrix $3A$ and find its determinant and equate it with the right-hand side to get the value of $K$.

If a row of a matrix is multiplied by a scalar quantity $y$ then its determinant also gets multiplied by the same scalar quantity.

Complete step by step solution:

Here we are given $A$ is \[{\text{3}} \times {\text{3}}\] matrix which implies it has $3$ rows.

We first need to find the matrix $3A$ to get the desired answer. So the matrix $3A$ will also have $3$ rows.

The matrix $3A$ can be obtained by multiplying each row of matrix $A$ by $3$.

Thus if $A$ has row vectors as $a_1,a_2,a_3$ then

The matrix $3A$ will have row vectors as \[3a_1,3a_2,3a_3\].

Since if a row of a matrix is multiplied by a scalar quantity $y$ then its determinant also gets multiplied by the same scalar quantity.

Hence if $B$ is any \[{\text{3}} \times {\text{3}}\] matrix whose each row gets multiplied by a scalar quantity y then

\[{\text{|yB|}} = {y \times y \times y|B|} \]

\[{\text{|yB|}} = {\text{y}}^{\text{3}}{\text{|B|}} \]

Now applying this formula for the given $3 \times 3$ matrix $A$ we get,

\[ {\text{|3A|}} = 3 \times 3 \times 3|A|\]  

\[{\text{|3A|}} = {\text{3}}^{\text{3}}{\text{|A|}} \]

\[{\text{|3A| = 27|A|}}......................\left( 1 \right) \]

Equating the RHS of given equation with RHS of equation (1) we get:

\[{\text{27|A| = K|A|}} \]

\[{\text{K = 27}} \]

Hence the value of K is 27. Therefore option D is the correct option.

Additional information:

A matrix is a rectangular array of numbers, expressions, or symbols. It is more precisely the arrangement of the data into rows and columns.

Scalar multiplication is the multiplication of a matrix and a scalar number.

Scalar multiplication is distributive i.e.

\[c\left( {A + B} \right) = cA + cB\]

Scalar multiplication is also associative i.e.

\[cd\left( A \right) = c\left( {dA} \right)\]

Note:

An Alternative method or direct trick to solve this question is to use the following formula:

If A is a \[{{n \times n}}\] matrix then,

\[{\text{|nA| = }}{{\text{n}}^{\text{n}}}{\text{|A|}}\]