Question

What is the determinant of the coefficient matrix of the system
–11
–2
0
55

Answer 1

The determinant of coefficient matrix of the given system is 0.

Explanation:

Given system of equation

`-x-y-z=-3`

`-x-y-z=8`

`3x+2y+z=0`

The coefficient matrix of the system

`\left[\begin{array}{ccc}-1&-1&-1\\-1&-1&-1\\3&2&1\end{array}\right]`

Let A=
`\left[\begin{array}{ccc}-1&-1&-1\\-1&-1&-1\\3&2&1\end{array}\right]`

X=
`\left[\begin{array}{}x&y&z\end{array}\right]`

B=
`\left[\begin{array}-3 &8&0\end{array}\right]`

Therefore , we can AX=B

`\left[\begin{array}{ccc}-1&-1&-1\\1&-1&-1\\3&2&1\end{array}\right]`
`\left[\begin{array}{}x&y&z \end{array}\right]`=
`\left[\begin{array}{}-3&8&0\end{array}\right]`

The determinant of coefficient matrix of the given system is given by

`\begin{vmatrix}-1&-1&-1\\-1&-1&-1\\3&2&1\end{vmatrix}`

By using determinant property : when two rowsor two columns are identical then the value of determinant is equal to zero .

`\therefore \begin{vmatrix}A\end{vmatrix}=0`

Answer 2

The first two rows of coefficients are identical, so by inspection, the determinant is 0.