Expansion by Minors

The determinant of matrices larger than 2*2 is calculated by expansion by minors. In this method, the matrix is repeatedly broken down into 2*2 matrices, and the determinants of each are combined to find the overall determinant.

Minor

Cofactors

Expansion by Minors of 3*3 Matrices

The first step to expand by minors is to choose any column or row to expand by. The row or column with the most zeroes and/or the smallest values will make the calculations easier, but any one will provide the correct answer.

The formula that expresses the determinant of a 3*3 matrix M using expansion by minors is:

Each element in the chosen row/column is multiplied by its cofactor. The sum of these products is the determinant. Any element that is zero can be skipped because the product of 0 and any number is also 0.

Examples

Find

Explanation

The first row will be the easiest to expand by because it contains a zero.

Remember that any row or column can be used with expansion by minors. The intermediate calculations will be different depending on which row or column is chosen, but the determinant will be the same regardless.

Try these exercises:

Solve

1. 


2. 


3. For every element of a 3*3 matrix, determine the power of -1 that would be used to calculate the element’s cofactor.


4. Find the determinant of N.
   


5. Find the determinant.
   


6. Find the determinant.
   


7. Find the determinant.
   


8. Find the determinant of I₃


9. Find the determinant.
   


10. Without carrying out any calculations, create a 3*3 matrix whose determinant is zero, with no more than 3 of the elements equal to zero.

Answers to questions

1. 


2. 


3. 


4. 


5. 0


6. -4


7. -360


8. 1


9. 0


10. Because the determinant of a 3*3 matrix can be determined by expanding by any row or column, any 3*3 matrix that has an entire row or column of zeroes provides the solution. For example, the determinant of both
    
    and
    
    is zero.