# - Find the determinant of a 5 X 5 matrix -

• VinnyCee
= (-1)\,(2)\,det\left(\left[\begin{array}{cccc}3 & 4 & 5 & 6 \\2 & 2 & 3 & 4 \\0 & 0 & 6 & 5 \\0 & 0 & 5 & 6\end{array}\right]\right)\,+\,(-1)\,(1)\,det\left(\left[\begin{array}{cccc}1 & 3 & 4 & 5 \\3 & 0 & 4 & 5 \\0 & 0 & 6 & 5 \\0 & 0 & 5 & 6
VinnyCee
--- Find the determinant of a 5 X 5 matrix ---

## Homework Statement

Find the determinant of the matrix.

A = $$\left[\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 3 & 0 & 4 & 5 & 6 \\ 2 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 6 & 5 \\ 0 & 0 & 0 & 5 & 6 \end{array}\right]$$

## Homework Equations

Laplace Expansion forumla

For an Expansion across the $$i^{th}$$ row of an n x n matrix:

det(A) = $$\sum_{j=1}^{n}\,(-1)^{i\,+\,j}\,a_{i\,j}\,det\left(A_{i\,j}\right)$$
(for a fixed i)

For an Expansion across the $$j^{th}$$ column of an n x n matrix:

det(A) = $$\sum_{i=1}^{n}\,(-1)^{i\,+\,j}\,a_{i\,j}\,det\left(A_{i\,j}\right)$$
(for a fixed j)

## The Attempt at a Solution

So, I start by doing a Laplace Expansion across the first row and down the second column. So i = 1 and j = 2.

det(A) = $$(-1)^{1\,+\,2}\,a_{1\,2}\,det\left(A_{1\,2}\right)$$

det(A) = $$(-1)\,(2)\,det\left(\left[\begin{array}{cccc} 3 & 4 & 5 & 6 \\ 2 & 2 & 3 & 4 \\ 0 & 0 & 6 & 5 \\ 0 & 0 & 5 & 6 \end{array}\right]\right)$$

I continue by doing another Laplace Expansion, this time across the first row and down the first column. So i = 1 and j = 1.

det(A) = $$(-2)\,(-1)^{1\,+\,1}\,a_{1\,1}\,det\left(\left[\begin{array}{ccc} 2 & 3 & 4 \\ 0 & 6 & 5 \\ 0 & 5 & 6 \end{array}\right]\right)$$

det(A) = $$(-2)\,(1)\,(3)\,det\left(\left[\begin{array}{ccc} 2 & 3 & 4 \\ 0 & 6 & 5 \\ 0 & 5 & 6 \end{array}\right]\right)$$

For the 3 x 3 matrix, I use Sarrus's Rule to get a determinant of 22.

det(A) = $$(-6)\,(22)\,=-132$$

However, when I plug the original matrix into my TI-92, I get det(A) = 99!

I tried a second time with i = 1 and j = 1 for the original matrix and then i = 1 and j = 2 for the 4 x 4 matrix. Here I get det(A) = -44.

Neither are right! What am I doing wrong here?

You haven't summed. Fix a row and you need to do the Laplace thing for every entry in that row, not just one that takes your fancy, and take the signed sum. In short, you've not used the formula properly.

It is easier to use row operations, anyway.

I found my error, it was exactly as you said. I was not doing the sum for the second non-zero term in the column!

The right equation is:

det(A) = $$(-1)^{1\,+\,2}\,a_{1\,2}\,det\left(A_{1\,2}\right)\,+\,(-1)^{3\,+\,2}\,a_{3\,2}\,det\left(A_{3\,2}\right)$$

Last edited:

## 1. How do I find the determinant of a 5 X 5 matrix?

To find the determinant of a 5 X 5 matrix, you can use the Laplace expansion method or the Gaussian elimination method. Both methods involve performing operations on the matrix to simplify it and eventually find the determinant.

## 2. Is there a shortcut to finding the determinant of a 5 X 5 matrix?

Unfortunately, there is no shortcut or formula for finding the determinant of a 5 X 5 matrix. The only way to find the determinant is to use one of the methods mentioned above.

## 3. What is the importance of finding the determinant of a 5 X 5 matrix?

The determinant of a matrix is a useful tool in linear algebra and is used to solve systems of linear equations, calculate the area and volume of geometric shapes, and determine whether a matrix is invertible. It also has applications in fields such as physics, engineering, and economics.

## 4. Can the determinant of a 5 X 5 matrix be negative?

Yes, the determinant of a 5 X 5 matrix can be negative. The determinant is a real number that can be positive, negative, or zero, depending on the values in the matrix.

## 5. Are there any properties or rules for finding the determinant of a 5 X 5 matrix?

Yes, there are several properties and rules that can make finding the determinant of a 5 X 5 matrix easier. These include the rule of triangle manipulation, the rule of multiplication, the rule of addition, and the rule of transpose. Knowing these properties can help simplify the process of finding the determinant.

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