Question

Find the determinant of the matrix. [20 0 10] [0 -30 0] [30 0 20] Determine whether the matrix has an inverse, but don't calculate the inverse. - The matrix has an inverse. - The matrix does not have an inverse

Answer 1

The determinant of the matrix is 3000, and since it is non-zero, the matrix has an inverse.

Step-by-step explanation:

To find the determinant of the given matrix and determine whether the matrix has an inverse, we use the following matrix:

[20 0 10]
[ 0 -30 0]
[30 0 20]

We can calculate the determinant of this 3x3 matrix using the rule of Sarrus or the standard method of minors and cofactors. Applying the standard formula for a 3x3 matrix:

1. Choose any row or column. Here, we'll pick the second row for convenience, as it contains two zeroes.
2. Expand the determinant across the chosen row or column.
3. For each element, calculate the determinant of the 2x2 submatrix that remains after removing the row and column of that element, multiplied by the element and its sign based on the element's position.

The calculation will look like this:

Determinant = (0)*[(0)(20) - (10)(0)] - (-30)*[(20)(20) - (10)(30)] + (0)*[(20)(0) - (0)(30)]
Determinant = 0 - (-30)*(400 - 300)+ 0
Determinant = 30*100
Determinant = 3000

Since the determinant is non-zero (3000), the matrix has an inverse.

To summarize, the determinant of the matrix is 3000, and because the determinant is not equal to zero, the matrix does indeed have an inverse.