4X4 Matrix determinant

In summary, the conversation is discussing a 4x4 matrix with determinant -8 and the question is what is the determinant of a new matrix obtained by performing row operations. The conversation includes a discussion on how row operations affect the determinant, such as swapping rows multiplying by a number, or adding rows to each other. The example provided uses row reduction to show that the determinant of the new matrix is also -8.
  • #1
186
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If a 4X4 matrix A with rows v1, v2, v3, and v4 has determinant det A = -8

then det
|(4v1+3v4)|
|___v2___|
|___v3___| =?
|(8v1+9v4)|


- I put the underscores for spacing.

The Attempt at a Solution


- I haven't attempting anything on this problem. There are no examples even remotely like this in the text and we haven't been shown anything like this in class, so I really have no clue how to even start. Please help.
 
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  • #2
Use "row reduction". Subtract 2 times the first row from the second row. Do you know what effect row reduction of a matrix has on a determinant.
 
  • #3
use row reduction on the 1X4 determinant given?
 
  • #4
There is no such thing as a "one by four" determinant! Every determinant, by definition, must be "square". There exist "one by four", or other dimension, matrices but only square matrices have determinants. You said yourself "a 4X4 matrix A with rows v1, v2, v3, and v4". The "rows" v1, v2, v3, and v4 must be 4 dimensional vectors. Subtracting twice the first row from the second row gives you a matrix with rows 8v1+ 3v4, v2, v3, 3v4. Subtracting the new third row from the first row gives a matrix with rows 8v1, v2, v3, 3v4. Dividing the first row by 8 gives a matrix with rows v1, v2, v3, 3v4. Finally, dividing the fourth rwo by 3 gives a matrix with rows v1, v2, v3, v4. How did those row operations change the value of the determinant of that matrix? If the final result is -8, what must the original determinant have been?
 
  • #5
Maybe I'm reading the way they have the determinant setup wrong. I was seeing it as 4v1+3v4 as row 1 and v2 as row 2 and v3 as row3 and and 8v1+9v4 as the 4th row (because I was viewing the determinant as a matrix I suppose). What is the correct way to read it? That might help a lot in me understanding the problem.
 
  • #6
This is where I'm confused. I can use row reduction on the new matrix and it is the same as the original (v1,v2,v3,v4), but I know the determinant isn't the same. How does that relate them?
 
  • #7
Yes, that is exactly right. And since v1, v2, v3, v4 have 4 members, this is a 4 by 4 determinant. For example, if v1= <1, 2, 0, 1> and v4= <-1, 0, 2, 1> are the top and bottom rows of the original matrix, then 4v1+ 3v4= <4, 8, 0, 4>+ <-3, 0, 6, 3>= <1, 8, 6, 7> and 8v1+ 9v4= <8, 16, 0, 8>+ <-9, 0, 18, 9>= <-1, 15, 18, 17> are the top and bottom rows of the new matrix, and of its determinant.

Here's one way to "cheat". Since the problem clearly expects a single answer, just set up a simple matrix, A, that has determinant -8.
[tex]A= \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 1\end{array}\right)[/tex]
where v1= <1, 0, 0, 0>, v2= <0, -2, 0, 0>, v3= <0, 0, 4, 0>, and v4= < 0, 0, 0, 1> will do nicely.
Now, 4v1+ 3v4= <4, 0, 0, 3> and 8v1+ 9v4= <8, 0, 0, 9>.

What is the determinant of
[tex]A= \left|\begin{array}{cccc} 4 & 0 & 0 & 3 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 4 & 0\\ 8 & 0 & 0 & 9\end{array}\right|[/tex]?

That's "cheating" because it assumes there is one single answer to this question and uses an example to find that one answer.
 
  • #8
You stuck in a message while I was responding to the previous one.

That's probably precisely what the problem is testing: your knowledge of how row operations affect the determinant of a matrix.

If you swap two rows in a matrix, you multiply the determinant by -1.

If you subtract a multiple of one row from another the determinant is not changed!

If you multiply an entire row by a number, the determinant is multiplied by that number.
 
  • #9
the determinant is -96 using the "cheating" method. Can you explain to me what row operations are being performed in this example? I'm trying to understand this before I move on to the vector space problems. Obviously rows are being multiplied by a number, but also being added to each other, so I'm not sure which rules that would apply to.
 

What is a 4x4 matrix determinant?

A 4x4 matrix determinant is a mathematical calculation used to determine the unique value of a 4x4 matrix. It involves multiplying and adding the elements of the matrix in a specific way to arrive at a single number.

Why is the 4x4 matrix determinant important?

The 4x4 matrix determinant is important in many areas of mathematics and science, including linear algebra, differential equations, and physics. It is also used in computer graphics and engineering applications.

How is the 4x4 matrix determinant calculated?

The 4x4 matrix determinant is calculated by expanding the matrix along any row or column and using the Laplace expansion method. This involves multiplying the elements of the chosen row or column by their corresponding minors (determinants of smaller submatrices) and then adding or subtracting these products according to a specific pattern.

What is the significance of the sign in the 4x4 matrix determinant calculation?

The sign in the 4x4 matrix determinant calculation alternates between positive and negative depending on the position of the element in the matrix. This is important because it ensures that the final result is not affected by the order of the elements in the matrix.

Can the 4x4 matrix determinant be used for larger matrices?

Yes, the 4x4 matrix determinant calculation can be extended to larger matrices, such as 5x5 or 6x6. However, the process becomes more complex and time-consuming as the size of the matrix increases.

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