Expanding by Minors: Understanding the Process of Computing Determinants

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    Determinant
In summary, a determinant is a mathematical value that can be calculated from a square matrix and represents various geometric and algebraic properties. To compute the determinant, multiple methods can be used such as the cofactor expansion, row reduction, or diagonalization. It has significant applications in mathematics, physics, and engineering and is used for solving equations, calculating areas and volumes, and determining invertibility. The determinant is also related to eigenvalues, as it is equal to the product of the eigenvalues and can be used to find them. Additionally, the determinant can be negative, indicating a change in orientation of the vectors in the matrix.
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brad sue
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Hi guys, what doe it mean when we are asked to compute the determinant the following way:

compute determinant across first row, down second column:
|1_0_3|
|2_2_1|
|4_0_3|

I know how to compute but here I need to understand the way to do it.

Thank you
B
 
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  • #2
They are likely asking you to compute the determinant in two ways:
(1) Across the first row
(2) Down the second column
As you know, you can expand the determinant across any row or column.
 
  • #3
It's talking about "expanding by minors". Do you know how to do that?
 

FAQ: Expanding by Minors: Understanding the Process of Computing Determinants

What is a determinant?

A determinant is a mathematical value that can be calculated from a square matrix. It represents a variety of geometric and algebraic properties of the matrix, such as the volume of a parallelepiped spanned by the column vectors of the matrix.

How do you compute the determinant of a matrix?

To compute the determinant of a matrix, you can use various methods such as the cofactor expansion method, the row reduction method, or the diagonalization method. These methods involve performing mathematical operations on the matrix to simplify it and eventually arrive at the determinant.

What is the significance of the determinant?

The determinant has several important applications in mathematics, physics, and engineering. It is used to solve systems of linear equations, calculate areas and volumes, determine whether a matrix is invertible, and more. It also has important interpretations in terms of transformations and change of variables.

What is the relationship between the determinant and eigenvalues?

The determinant of a matrix is equal to the product of its eigenvalues. This means that if you know the eigenvalues of a matrix, you can compute its determinant by simply multiplying them together. Similarly, if you know the determinant of a matrix, you can find the eigenvalues by factoring the determinant.

Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. In fact, the sign of the determinant is an important property that can tell us about the orientation of the vectors in the matrix. A negative determinant indicates that the matrix has been transformed in a way that flips the orientation of its vectors.

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