Evaluate the determinant of the matrix.

[-4 5 6
0 4 4
-2 -5 4]

To evaluate the determinant of a matrix, you can use the method of cofactor expansion along any row or column. Let's use the first column to find the determinant of the matrix:

Given matrix:
[-4 5 6]
[0 4 4]
[-2 -5 4]

Step 1: Start with the first element in the first column (-4).
Step 2: Multiply it by the determinant of the 2x2 submatrix formed by removing the row and column containing that element.
| 4 4 |
| -5 4 |
Step 3: Apply the alternating sign pattern (+ - +) based on the position of the element.
Step 4: Repeat the process for each element in the first column and add the results together to get the determinant.

Determinant = -4 * (4 * 4 - (-5 * 4)) + 5 * (0 * 4 - (-2 * 4)) + 6 * (0 * (-5) - (-2 * 4))

Simplifying further:

Determinant = -4 * (16 + 20) + 5 * (0 + 8) + 6 * (0 + 8)
Determinant = -4 * 36 + 5 * 8 + 6 * 8
Determinant = -144 + 40 + 48
Determinant = -144 + 88
Determinant = -56

Therefore, the determinant of the given matrix is -56.

To evaluate the determinant of a matrix, we can use different methods such as cofactor expansion, row operations, or using special properties of certain matrices.

Let's use the cofactor expansion method to evaluate the determinant of the given matrix:

The formula for calculating the determinant of a 3x3 matrix is as follows:

det(A) = a11 * C11 - a12 * C12 + a13 * C13

where aij represents the element in the ith row and jth column, and Cij represents the cofactor of the element aij.

The matrix A is:

A = [-4 5 6
0 4 4
-2 -5 4]

Let's calculate each cofactor one by one:

C11 = det([4 4
-5 4]) = (4*4) - (4*-5) = 16 + 20 = 36

C12 = det([0 4
-2 4]) = (0*4) - (4*-2) = 0 + 8 = 8

C13 = det([0 4
-2 -5]) = (0*(-5)) - (-2*4) = 0 + 8 = 8

Now, let's substitute the values into the formula:

det(A) = (-4 * 36) - (5 * 8) + (6 * 8) = -144 - 40 + 48 = -136

Therefore, the determinant of the given matrix is -136.