Question

Explain why$\left[\begin{array}{ccc}2& 1& 7\\ 3& -5& 0\end{array}\right]$doesn’t have a determinant.

Short answer

The matrix $\left[\begin{array}{ccc}2& 1& 7\\ 3& -5& 0\end{array}\right]$ is not a square matrix and so as per the definition of the determinant, the matrix will not have a determinant.

Step-by-step solution

- Define a determinant of a matrix

A determinant is a number obtained by solving a square matrix only by certain methods which tells us about certain properties of the respective matrix.

- Define a determinant of second order matrix

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$.

For example: Apply the definition of the method of finding determinant of second order matrix $\left|\begin{array}{cc}8& 3\\ -5& 2\end{array}\right|$.

$\begin{array}{c}\left|\begin{array}{cc}8& 3\\ -5& 2\end{array}\right|=8\times \left(2\right)-3\left(-5\right)\\ =16+15\\ =31\end{array}$

- State the conclusion

Since $\left[\begin{array}{ccc}2& 1& 7\\ 3& -5& 0\end{array}\right]$is the matrix of order $2\times 3$, therefore the given matrix is not a square matrix and so as per the definition of the determinant, the matrix will not have one.