I have to compute the determinant of the matrix without using a calculator. If the determinant is zero, write singular matrix.
1 -1 -1
-4 2 0
0 3 1
To compute the determinant of a 3x3 matrix without using a calculator, you can use the method called "cofactor expansion" or the "Laplace expansion". Here are the steps to find the determinant of the given matrix:
Step 1: Multiply each element of the first row by its corresponding cofactor, which is found by alternating signs (+ and -) and multiplying the minor determinant (the determinant of the 2x2 matrix formed by excluding the current row and column) of the element. Let's call this expanded formula "Expansion 1".
Expansion 1:
det = 1 * det(2 0; 3 1) - (-1) * det(-4 0; 0 1) - (-1) * det(-4 2; 0 3)
Step 2: Compute the determinant of each 2x2 matrix using the same method. Let's call these determinants "Minor 1", "Minor 2", and "Minor 3".
Minor 1:
det(2 0; 3 1) = (2 * 1) - (0 * 3) = 2
Minor 2:
det(-4 0; 0 1) = (-4 * 1) - (0 * 0) = -4
Minor 3:
det(-4 2; 0 3) = (-4 * 3) - (2 * 0) = -12
Step 3: Substitute the computed determinant values back into Expansion 1.
det = 1 * 2 - (-1) * (-4) - (-1) * (-12)
det = 2 + 4 + 12
det = 18
The determinant of the given matrix is 18, which is non-zero. Therefore, the matrix is not singular.