Linear algebra: determinants (proof)

In summary, the conversation discusses proving that the determinant of a matrix M, which is composed of a kxk matrix A, a pxp matrix C, and a zero matrix O, is equal to the determinant of A multiplied by the determinant of C. The individual provides their attempt at a proof, but realizes their mistake in treating the matrices as scalars. They thank for the help in pointing out their error.
  • #1
yoda05378
11
0
hi, i seem to have some trouble proving:

Suppose M = [A B:O C], where A is a kxk matrix, C is a pxp matrix, and O is a zero matrix. Show that det(M) = det(A)det(C).


my attempt at a proof:

det(M) = det(A)det(C)

det[A B:O C] = det(A)det(C)

AC - OB = det(A)det(C)

AC = det(A)det(C) <-- doesn't make sense!

please point out where my logic failed.
 
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  • #2
nevermind. i figured out where i went wrong (i treated the matrices as if they were scalar, stupid me). thanks for all the help :sarcasm:
 

1. What is a determinant in linear algebra?

A determinant is a mathematical value that is calculated from a square matrix. It is used to determine various properties of a matrix, such as whether it is invertible or singular.

2. How is the determinant of a matrix calculated?

The determinant of a matrix can be calculated using various methods, such as the cofactor expansion method or using row operations to transform the matrix into a triangular form. The exact method used depends on the size and complexity of the matrix.

3. What is the significance of the determinant in linear algebra?

The determinant is an important concept in linear algebra as it helps determine various properties of a matrix, such as invertibility and linear independence. It is also used in solving systems of linear equations and finding eigenvalues and eigenvectors.

4. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the number of row swaps performed during the calculation. A negative determinant indicates that the matrix has an odd number of row swaps.

5. Is there a geometric interpretation of the determinant?

Yes, the determinant has a geometric interpretation as the factor by which a linear transformation changes the volume of a vector space. It can also be thought of as the area or volume of the parallelepiped formed by the column vectors of the matrix.

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