Find the following determinants

In summary, to find the determinant of a matrix, you can use the cofactor formula or perform elimination to reduce the matrix to a smaller one. The general pattern is to multiply each entry from the first row by the determinant of the matrix with that entry's row and column removed, and alternate the signs with (-1)^(row+col). This process can be repeated until you reach a 1x1 matrix, which is just the value of the remaining entry.
  • #1
adc85
35
0
I know how to find the determinant in general but these two problems here are tough for me:

1. Find the determinant of:
3 -1 0 0 0
-1 3 -1 0 0
0 -1 3 -1 0
0 0 -1 3 -1
0 0 0 -1 3

2. Find the determinant of:
0 2 2 2 2
2 0 2 2 2
2 2 0 2 2
2 2 2 0 2
2 2 2 2 0

Now, I know that I need to select a row or column that contains all zeros except for one number. The row or column that the one non-zero number is on would be selected as well. Then you would use the formula (for selecting row i and column j):

det A = (-1)^(i + j) * det A (without row i and column j)

But I can't figure out how to create a row or column full of zeros (except one element) for both of these problems. Thanks for any help.
 
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  • #2
adc85 said:
I know how to find the determinant in general but these two problems here are tough for me:

2. Find the determinant of:
0 2 2 2 2
2 0 2 2 2
2 2 0 2 2
2 2 2 0 2
2 2 2 2 0

Haven't done one of these for an age. How about this one. Subtract the row below from each row

-2 2 0 0 0
0 -2 2 0 0
0 0 -2 2 0
0 0 0 -2 2
2 2 2 2 0

Add the first 4, then the next 3 then the next 2

-2 0 0 0 2
0 -2 0 0 2
0 0 -2 0 2
0 0 0 -2 2
2 2 2 2 0

Add the first 4 to the last

-2 0 0 0 2
0 -2 0 0 2
0 0 -2 0 2
0 0 0 -2 2
0 0 0 0 8

Subtrace 1/4 of last from each of the others

-2 0 0 0 0
0 -2 0 0 0
0 0 -2 0 0
0 0 0 -2 0
0 0 0 0 8

Det = 128
 
  • #3
you can use cofactor formula or just do elimination like what OlderDan did. i prefer doing elimination and multiply diagonals to get determinant. i think that's how computer does it also. however on a test, you might have to write down the cofactors explicitly. i don't really remember all the details, but i believe the general idea is to reduce the problem from finding det of large matrix to a small one.
I'll illustrate w/ a smaller matrix
a b c
d e f
g h i

if you expand along the first row, it's
a * det(e f; h i) - b * det(d f; g i) + c * det(d e; g h)
so you can see, the general pattern is
(entry j from row 1) * (det of matrix w/ row 1, column j erased) * (-1)^(row+col)
i think the sign is built into cofactors, but from here i hope you can see its like reduction formula. i mean you can reduce it to finding det of 1x1 matrix.
det(e f; h i) = e * det(i) - f * det(h)
again the sign comes from (-1)^(row+col)
 

What is the purpose of finding determinants?

The purpose of finding determinants is to solve systems of linear equations, calculate areas and volumes, and determine the invertibility of a matrix.

What are the methods for finding determinants?

There are various methods for finding determinants, such as using the cofactor expansion method, the row reduction method, or the Laplace expansion method.

What are the properties of determinants?

Determinants have several properties, including the ability to be multiplied by a constant, to be added to another determinant, and to have their rows or columns switched without changing the value.

How do I calculate the determinant of a 3x3 matrix?

To calculate the determinant of a 3x3 matrix, you can use the Sarrus method, which involves multiplying certain elements of the matrix and adding them together in a specific pattern.

What are some real-world applications of finding determinants?

Finding determinants has various real-world applications, such as in engineering for solving systems of linear equations in circuit analysis, in physics for calculating moments of inertia, and in economics for solving input-output models.

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