Question

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

9. $\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 2& 2& 2\\ 1& 1& 3& 3& 3\\ 1& 1& 1& 4& 4\\ 1& 1& 1& 1& 5\end{array}\right]$

Short answer

Therefore, the determinant of given matrix is given by,

$\det \mathrm{A}=24$

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,

$A=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 2& 2& 2\\ 1& 1& 3& 3& 3\\ 1& 1& 1& 4& 4\\ 1& 1& 1& 1& 5\end{array}\right]$

To find determinant by using Gaussian Eliminations

We multiply the first row by -1 and add it to every other row.

We get,

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

We had zero row interchanges, so

$$\begin{gathered} \det \mathrm{A}=\left(-1\right)°.1.1.2.3.4 \\ \det \mathrm{A}=24. \end{gathered}$$