Question

If A is a $3 \times 3$ matrix, $\left| {3A} \right| = k\left| A \right|$ , then write the value of k.

Hint:Here, we can apply the fundamental property of the determinant value of the matrix. The required value of k can be obtained easily.

The given matrix is of $3 \times 3$ order. Thus,
n = 3.
For a matrix of order n, we know that,
$\left| {cA} \right| = {c^n}\left| A \right|$ , where c is a constant.
The above property is valid, because the number of rows in the matrix of order n will be n. So, cA will give a matrix where all elements will be multiplied by c. Due to n rows, the determinant of matrix cA will be effectively multiplied by ${c^n}$ .

Since $\left| {cA} \right| = {c^n}\left| A \right|$ where n is the order of the matrix.
So, substitute c=3 and n=3 in above equation, we get,
$\left| {3A} \right| = {3^3}\left| A \right| \\ \Rightarrow \left| {3A} \right| = 27\left| A \right| \\$ ….(1)
Also, it is given that $\left| {3A} \right| = k\left| A \right|$ ….(2)
Thus comparing equations (1) and (2), we will have
k=27.
$\therefore$ The value of k will be 27.

Note:The determinant is a scalar value that can be computed from the elements of a square matrix, and which encodes certain properties of the linear transformation described by the matrix.
Some properties of determinant are:
1.The value of the determinant does not change, if both rows and columns are interchanged.
2.If any two rows or columns are interchanged, then the sign of it changes.
3.If any two rows or columns are identical then the determinant will have zero value.
4.If each element of a row or a column is multiplied by a constant c, then its value gets multiplied by c.
5.If some or all elements of a row or column are expressed as the sum of two or more) terms, then the determinant can be expressed as the sum of two or more determinants.