Question

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

1.$\left[\begin{array}{ccc}1& 1& 1\\ 1& 3& 3\\ 2& 2& 5\end{array}\right]$

Short answer

Therefore, the determinant of given matrix is given by,

$\det A=6$

Step-by-step solution

Definition

Gaussian elimination method is used to solve a system of linear equations.

Gaussian elimination provides a relatively efficient way of constructing the inverse to a matrix.

Interchanging two rows, multiplying a row by a constant (any constant which is not zero).

Given

Given Matrix,

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 3& 3\\ 2& 2& 5\end{array}\right]$

To find determinant by using Gaussian Eliminations

First, we multiply the first row by -1 and add it to the second row. Then, we multiply the first row by -2 and add it to the third row.

We get,

$A_1=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right]$

We had zero row swaps, so

$$\begin{gathered} \mathrm{detA}={(-1)}^0\left|{\mathrm{A}}_1\right| \\ ={(-1)}^0.1.2.3 \\ =6 \end{gathered}$$