Find the determinants for the matrices:

Question

1. [ 5 4 ]

[ 1 6 ]

2. [ 1 5 -3 ]

[ -1 -3 3 ]

[ 0 4 4 ]

3. 3I (I is a 4x4 matrix)

4. [ -1 -3 0 -3 ]

[ 0 0 -1 0 ]

[ 1 -1 3 -2 ]

[ 3 2 -3 4 ]

Answer 1

The first one, being a 2x2 matrix, can be calculated using the formula ad - bc, where a = 5, b =4, c = 1, and d =6. With matrices, you always want to multiply diagonally moving from the left to the right.

(5*6)-(4*1) = 30-4 = 26

For a 3 by 3 matrix, you want to look at the top row separate from the bottom two rows. Now you want to multiply each number in the top row by the determinant of the matrix formed by the 4 numbers not in the same column as the number in the top row. For example, with number 2, you want to start with multiplying your first number in the top row, 1, by the determinant of the 2x2 matrix formed by the 4 numbers in the bottom 2 rows and not in 1's column.

Determinant of the 2x2 = (-3*4) - (3*4) = -12 -12 = -24

You now want to multiply the -24 by 1 (the first number in the first row) , giving you -24

Now do the same with the rest of the numbers in the first row. This gives you the following:

5*((-1*4)-(3*0)) = 5*(-4-0) = -20

-3*((-1*4)-(-3*0)) = -3*(-4-0) = 12

Now simply add the 3 to get your final answer. -24 + -20 + 12 = -32

3) Not sure what 3I means, but the same pattern continues for 4x4 matrices as well, as demonstrated in the next problem

4) For 4x4 matrices, the same pattern as the 3x3 matrices continues, with each number in the top row being multiplied with the determinant of the 3x3 matrix not in that column. These products are then added up to give you the determinant.

Hope this helps!