Question

If A is a square matrix of order 3 and \[\left| {\rm{A}} \right| = 4\], then find the value of \[\left| {{\rm{2A}}} \right|\].

Answer 1

Hint:
Here, we need to find the value of \[\left| {{\rm{2A}}} \right|\] by using the property of determinants.
We will use the order of the matrix and properties of determinant to rewrite the expression. Then we will substitute the given value of the determinant of A in the obtained expression to find the required answer.

Complete step by step solution:
We will use the property of determinants to find the value of \[\left| {{\rm{2A}}} \right|\].
If \[{\rm{A}} = \left[ {{a_{ij}}} \right]\] is a square matrix of order \[n\], and \[\left| {\rm{A}} \right|\] is the determinant of matrix A, then the determinant of matrix A multiplied by a scalar quantity \[k\], is equal to the product of determinant of A, and the scalar quantity \[k\] raised to the power \[n\]. This can be written as \[\left| {k{\rm{A}}} \right| = {k^n}\left| {\rm{A}} \right|\].
The matrix A is a square matrix of order 3.
Thus, we get
\[n = 3\]
We need to find determinant of matrix A multiplied by scalar quantity 2.
Thus, we get
\[k = 2\]
Now, substituting \[n = 3\] and \[k = 2\] in the property of determinant, we get
\[ \Rightarrow \left| {2{\rm{A}}} \right| = {2^3}\left| {\rm{A}} \right|\]
We know that the cube of 2 is 8.
Therefore, we get
\[ \Rightarrow \left| {2{\rm{A}}} \right| = 8\left| {\rm{A}} \right|\]
It is given that the determinant of matrix A is equal to 4.
Thus, substituting \[\left| {\rm{A}} \right| = 4\] in the equation, we get
\[ \Rightarrow \left| {2{\rm{A}}} \right| = 8 \times 4\]
The product of 8 and 4 is 32.
Therefore, multiplying the terms in the expression, we get
\[ \Rightarrow \left| {2{\rm{A}}} \right| = 32\]

Thus, we get the value of \[\left| {{\rm{2A}}} \right|\] as 32.

Note:
The matrix given in the question is a square matrix. A square matrix is a matrix whose number of rows is equal to the number of columns. The determinant of a matrix exists only if the matrix is a square matrix. A determinant is a number that any square matrix can be associated with.
Here we will first covert the expression \[\left| {{\rm{2A}}} \right|\] in terms of \[\left| {\rm{A}} \right|\], so that we can easily substitute the value of \[\left| {\rm{A}} \right|\] and find the required answer.