Question

29. If$A=\left[\overrightarrow{u}\overrightarrow{v}\overrightarrow{w}\right]$is a$3\times 3$ matrix, then the formula$det\left(A\right)=\overrightarrow{V}.\left(\overrightarrow{u}\times \overrightarrow{w}\right)$ must hold.

Short answer

Therefore, the given statement is True.

Step-by-step solution

Matrix Definition. 

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “ m by n ” matrix, written “ $m\times n$.”

To check whether the given condition is true or false.

Let,

For example,

$A=l_3,$

then

$\begin{array}{l}u=e_1,\\ v=e_2,\\ w=e_3,\end{array}$

Now,

$$\begin{gathered} v.\left(u\times w\right)=e_2.\left(e_1\times e_3\right) \\ v.\left(u\times w\right)=e_2.\left(-e_2\right) \\ v.\left(u\times w\right)=-1 \\ v.\left(u\times w\right)\ne 1 \\ v.\left(u\times w\right)=detA \end{gathered}$$

Therefore, the given statement is True.