Question

What is the determinant of the coefficient matrix of the system

0
3
6
7

Answer 1

Imma go with 0 for the answer.

Answer 2

D=0

Explanation:

First we are going to express the system in matrix form:

-3x+0y-2z=6

9x+0y+5z=7

6x+0y-12z=3

Expressing this in a matrix is:

`\left[\begin{array}{cccc}-3&0&-2&|6\\9&0&5&|7\\6&0&-12&|3\end{array}\right]`

But in this case we just need to calculate the determinant, then the matrix is:

`\left[\begin{array}{ccc}-3&0&-2\\9&0&5\\6&0&-12\end{array}\right]`

There's a property that says that a square matrix with a row or a column in which all elements are null has a determinant equal to zero.

And we can see that in this case this matrix has a row which all elements are null. Then the determinant of this matrix is null.

But, we're going to calculate it anyways, to check this property.

We have by definition of determinant:

`\left[\begin{array}{ccc}-3&0&-2\\9&0&5\\6&0&-12\end{array}\right]\\\\D=(-3)(-1)^1^+^1\left[\begin{array}{cc}0&5\\0&-12\end{array}\right] +0(-1)^1^+^2\left[\begin{array}{cc}9&5\\6&-12\end{array}\right]+(-2)(-1)^1^+^3\left[\begin{array}{cc}9&0\\6&0\end{array}\right`]

`D=(-3).(0.(-12)-0.5)+0+(-2).(9.0-6.0)\\=(-3).0+0+(-2).0\\=0`

Then we got that the determinant of the coefficient matrix of the system is D=0.