When is Matrix C Not Invertible?

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In summary, the conversation discusses the determinant of a 3x3 matrix, denoted as C, and how it relates to the values of 'x' in the matrix. The determinant of C is calculated to be x^2 - 12x + 27. When multiplied by a constant, the determinant is also multiplied by that constant, as seen in the calculation for det(2C) = 2^3 det(C). The values of 'x' that make C non-invertible are determined by setting the determinant to 0, resulting in the solutions x = 9 or 3. The conversation also mentions the trace of a matrix, which is related to the determinant but not the same.
  • #1
dcl
55
0
[tex]
c = \left[ {\begin{array}{*{20}c}
{2 - x} & 5 & 1 \\
{ - 3} & 0 & x \\
{ - 2} & 1 & 2 \\
\end{array}} \right][/tex]

a) Calculate det(C).
My answer was [tex]x^2 - 12x + 27[/tex].

b) Calculate det(2C).
Umm, would this just be 2*det(C)?
Couldn't find anything more helpful in my notes.

c) State the values for 'x' for which C is not invertible.
I believe the value for 'x' that would make this non invertable would be the solution that det(c) = 0. (A matrix has no inverse when the determinant = 0 yeh?)
which would be x = 9 or 3
is this correct?
 
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  • #2
I get the same answer as you on (a) and (c). I'm thinking that det(aC)=a^N det(c) where a is a constant and N is the "dimension" (wrong terminology? NxN matrix). If I am right about that, then for a 3x3 matrix, det(2C) = 8 det(C).

You may be thinking about the trace of a matrix: Tr(2C) = 2 Tr(C).
 
Last edited:
  • #3
Hmm, yeh, what you're saying would make more sense.
Thanks for confirming the other answers.
 
  • #4
Janitor is correct about (b). To see it, just consider what happens if C is the identity: 2I has 2 along the diagonal and 0 everywhere else, so the determinant is det(2I) = 8 = 23det(I).
 

FAQ: When is Matrix C Not Invertible?

What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is used to represent and manipulate data or equations in a concise and organized manner.

What is a determinant?

A determinant is a numerical value that can be calculated from a square matrix. It represents certain properties of the matrix, such as the areas and volumes of geometric figures represented by the matrix.

How do you find the determinant of a matrix?

The determinant of a matrix can be found by using a specific formula depending on the size of the matrix. For a 2x2 matrix, the determinant is calculated by multiplying the top left and bottom right elements and subtracting the product of the top right and bottom left elements. For larger matrices, various methods such as expansion by minors or using row operations can be used to calculate the determinant.

What is an inverse matrix?

An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. In other words, it "undoes" the effects of the original matrix. It is denoted by adding a superscript -1 to the original matrix, and can be found using various methods such as Gaussian elimination or using the adjugate matrix.

Why are matrices and their properties important in science?

Matrices and their properties are important in science because they provide a powerful tool for representing and manipulating data and equations. They are used in various fields such as physics, computer science, and engineering, to solve complex problems and make predictions. Additionally, the properties of matrices, such as determinants and inverses, have practical applications in areas like statistics and machine learning.

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