Question

# Find the determinants in Exercise , assuming that begin{vmatrix}a & b &c d&e&fg&h&i end{vmatrix}=4 begin{vmatrix}2a & 2b &2c d&e&fg&h&i end{vmatrix}

Matrices
Find the determinants in Exercise , assuming that
$$\begin{vmatrix}a & b &c \\ d&e&f\\g&h&i \end{vmatrix}=4$$
$$\begin{vmatrix}2a & 2b &2c \\ d&e&f\\g&h&i \end{vmatrix}$$

2020-12-08
Step 1
Determinant is used for solving linear equations. To find the inverse of a matrix determinant is used. It is computed only for square matrices. Square matrices are matrices whose number of rows and number of columns are equal.
Step 2
The given value of the determinant is $$\begin{vmatrix}a & b &c \\ d&e&f\\g&h&i \end{vmatrix}=4$$ Multiplying any row or column of a determinant by a number x, multiplies the determinant by that number x. Using this property the value of the determinant $$\begin{vmatrix}2a & 2b &2c \\ d&e&f\\g&h&i \end{vmatrix}$$ is calculated by taking the common term 2 from the first row and take the value 2 outside of the determinant, then it becomes 2 times of the determinant $$\begin{vmatrix}a & b &c \\ d&e&f\\g&h&i \end{vmatrix}$$ , using the value given value find the asked determinant value as follows,
$$\begin{vmatrix}a & b &c \\ d&e&f\\g&h&i \end{vmatrix}=4$$
$$\begin{vmatrix}2a & 2b &2c \\ d&e&f\\g&h&i \end{vmatrix}=\begin{vmatrix}2 \times a & 2 \times b &2 \times c \\ d & e & f \\ g & h &i \end{vmatrix}$$
$$=2\begin{vmatrix}a & b &c \\ d&e&f\\g&h&i \end{vmatrix}$$ Take the coomon term from first row outside of determinant. $$= 2 \times 4 \dotsc \text{ Subtitute } \begin{vmatrix}a & b &c \\ d&e&f\\g&h&i \end{vmatrix}=4$$
$$= 8$$
Hence, the value of the determinant asked in the question is equal to
$$\begin{vmatrix}2a & 2b &2c \\ d&e&f\\g&h&i \end{vmatrix} = 8$$