Chapter 10.6, Problem 21E

To determine

The determinant of the given matrix is to be calculated and tell that does it contain the inverse matrix or not:

  $\left[\begin{array}{ccc}30& 0& 20\\ 0& -10& -20\\ 40& 0& 10\end{array}\right]$

Answer to Problem 21E

The determinant of $\left[\begin{array}{ccc}30& 0& 20\\ 0& -10& -20\\ 40& 0& 10\end{array}\right]$ is $5000$ and it contains the inverse matrix.

Explanation of Solution

Given:

  $\left[\begin{array}{ccc}30& 0& 20\\ 0& -10& -20\\ 40& 0& 10\end{array}\right]$

Concept Used:

The determinant of every square matrix with the order $n$ is defined as $A=\left[a_{ij}\right]$ where,

  $a={\left(i,j\right)}^{th}$ of element $A$ .

The given matrix has the inverse matrix when:

• A matrix must contain the same number of rows and column
• its determinant value must not be zero value

Calculations:

The determinant of the given matrix is calculated as:

  $\begin{array}{l}D=\left[\begin{array}{ccc}30& 0& 20\\ 0& -10& -20\\ 40& 0& 10\end{array}\right]\\ D=30\left[\left(-10\times 10\right)-\left(0\times -20\right)\right]-0\left[\left(0\times 10\right)-\left(40\times -20\right)\right]+20\left[\left(0\times 0\right)-\left(40\times -10\right)\right]\\ D=30\left(-100\right)-0+20\left(400\right)\\ D=-3000+8000\\ D=5000\end{array}$

Now, the given matrix is $\left(3\times 3\right)$ and its determinant value is $D=5000$ or non-zero value which clearly shows that this matrix contains the inverse matrix.

Conclusion:

Hence, the determinant of the given matrix is calculated as $D=5000$ .