Finding determinant given determinant of another matrix

In summary, to find the determinant of matrix A satisfying A^2-3A-2I=0, where I is the identity matrix, given that the determinant of A-3 is 2, you can use the property det(AB)=det(A)det(B) and factor A^2-3A to get A(A-3)=2I. Then, using the property again, you can solve for det(A) as det(A)=det(2I)/det(A-3).
  • #1
bojo
5
0

Homework Statement



Let A be a 3 x 3 matrix satisfying the equation [tex]A^{2}[/tex]-3A-2I=0 where I is the 3x3 identity matrix. Find det(A) given the det(A-3)=2

The Attempt at a Solution



Well can't find anything like this in my textbook, notes or google. I imagine its a pretty simple matrix property I've overlooked but otherwise i have no clue what to do!


Help much appreciated,
Ben
 
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  • #2
det(AB) =det(A)*det(B)

So if you factor A^2-3A you can use det(A-3) = 2
 
  • #3
so

A^2 -3A = 2I factors to

A(A-3)=2I

and using property det(AB)=det(A)det(B)

det(A)det(A-3)=det(2I) so det(A)=det(2I)/det(A-3)
 
  • #4
Yes and you can easily find det(2I).
 

What is a determinant?

A determinant is a mathematical quantity that is calculated for a square matrix. It is used to determine various properties of the matrix, such as its invertibility and the volume of the parallelepiped spanned by the matrix's columns.

How is the determinant calculated?

The determinant of a matrix can be calculated using various methods, such as cofactor expansion, row reduction, or using the properties of determinants. The specific method used depends on the size of the matrix and the available information.

What information is needed to find the determinant of a matrix?

The determinant of a matrix can be found using the values of the matrix's entries. For larger matrices, the values of the submatrices formed by removing one row and one column at a time may also be needed.

Can the determinant of one matrix be used to find the determinant of another matrix?

Yes, the determinant of one matrix can be used to find the determinant of another matrix, as long as the two matrices are related in some way. For example, if one matrix is the transpose of the other, their determinants will be equal.

What are some applications of finding the determinant of a matrix?

Finding the determinant of a matrix is useful in various areas of mathematics and science, such as solving systems of linear equations, calculating the inverse of a matrix, and determining the eigenvalues of a matrix.

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