Question

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

3.$\left[\begin{array}{cccc}1& 3& 2& 4\\ 1& 6& 4& 8\\ 1& 3& 0& 0\\ 2& 6& 4& 12\end{array}\right]$

Short answer

Therefore, the determinant of given matrix is given by,

$\det A=-24$

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,

$A=\left[\begin{array}{cccc}1& 3& 2& 4\\ 1& 6& 4& 8\\ 1& 3& 0& 0\\ 2& 6& 4& 12\end{array}\right]$

To find determinant by using Gaussian Eliminations

First, we multiply the first row by -1 and add it to the second and third row. Then, we multiply the first row by -2 and add it to the 4th row.

We get,

$A=\left[\begin{array}{cccc}1& 3& 2& 4\\ 0& 3& 2& 4\\ 0& 0& -2& -4\\ 0& 0& 0& 4\end{array}\right]$

We had zero row swaps, so

$\begin{array}{l}detA={\left(-1\right)}^0.1.3.-2.4\\ detA=-24\end{array}$