E1.4b Determinant with zero column

In summary, the provided conversation discusses the process of evaluating a determinant of a matrix by using row operations. It is mentioned that if two rows or columns of a matrix are equal, then the determinant is 0. The conversation also mentions that the co-factor expansion would result in multiplying zeros throughout. However, it is suggested that by using row operations, the matrix can be reduced to a simpler form, which can make it easier to evaluate the determinant. It is also noted that if two rows or columns of a matrix are the same, the determinant will be 0.
  • #1
karush
Gold Member
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$$\left[\begin{array}{rrrrr}
1 &0 &2 &1\\
1 &1 &0 &1\\
1 &3 &4 &1\\
-1 &-3 &-4 &-1
\end{array}\right]=\color{red}{0}$$Answer (red) via W|Aok I did not do any operations on this
Since by observation the 4th column can become all zero'showever didn't see anything in the book to support this
only that the co-factor expansion would result in multiplying zeros throughoutany suggestions?
 
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  • #2
Surely in your textbook there is the rule "If two rows or columns of a matrix are equal" (or the more general "if one row or column is a multiple of another") "then the determinant is 0". One way to evaluate the determinant of matrix is to use "row operations" to reduce the matrix to a simpler matrix. The row operations are "multiply an entire row by a constant", "swap two rows", and "add a multiple of on row to another". The first multiplies the determinant by that constant. The second multiplies the determinant by -1. The third does not change the determinant.

In particular, if two rows of a matrix are the same, adding -1 times one of those rows to the other gives a matrix having all "0"s in one row. "Expanding" on that row gives 0 as the determinant 0.

(Of course you can replace "row" by "column" thoughout.)
 

FAQ: E1.4b Determinant with zero column

1. What is a determinant with zero column?

A determinant with zero column is a square matrix with at least one column (vertical) that contains all zeros. This means that the determinant of the matrix will also be equal to zero.

2. How is the determinant of a matrix with a zero column calculated?

The determinant of a matrix with a zero column is calculated by expanding along the column with zeros. This means that the determinant will be equal to the sum of the products of the elements in the zero column and their corresponding cofactors.

3. Can a matrix with a zero column have a non-zero determinant?

No, a matrix with a zero column will always have a determinant of zero. This is because the determinant is calculated by multiplying the elements in each row by their corresponding cofactors, and since the zero column contains all zeros, the product will always be zero.

4. What does a zero column in a matrix represent?

A zero column in a matrix represents a linearly dependent set of vectors. This means that one or more of the vectors in the matrix can be written as a linear combination of the other vectors in the matrix.

5. How does a zero column affect the invertibility of a matrix?

A zero column in a matrix makes the matrix non-invertible. This is because the determinant of the matrix is equal to zero, and for a matrix to be invertible, its determinant must be non-zero.

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