Evaluate the determinant of the matrix M = ⎡⎣⎢32−14136−15⎤⎦⎥

To evaluate the determinant of a 3x3 matrix, we use the formula given by:

det(M) = a(ei − fh) − b(di − fg) + c(dh − eg)

For the matrix M:

M =  ⎡ ⎣ ⎢
 3   2   −1
 4   1   3
 6   −1   5
⎤ ⎦ ⎥

The matrix is structured as:

• a = 3, b = 2, c = -1
• d = 4, e = 1, f = 3
• g = 6, h = -1, i = 5

Let's calculate each component separately:

• ei − fh = (1 * 5) - (3 * -1) = 5 + 3 = 8
• di − fg = (4 * 5) - (3 * 6) = 20 - 18 = 2
• dh − eg = (4 * -1) - (1 * 6) = -4 - 6 = -10

Now, plug these values back into the determinant formula:

det(M) = 3(8) - 2(2) + (-1)(-10)

det(M) = 24 - 4 + 10

So, det(M) = 30

The determinant of the matrix M is 30.