# 311.3.2.16 Find the determinant with variables a b c d e f g h i

• MHB
• karush
In summary: So there are multiple ways to row reduce a matrix to RREF, but only one determinant.In summary, when finding the determinants of a matrix, it is important to consider the use of complex numbers and the effect of row operations on the determinant. While there may be multiple ways to row reduce a matrix to reduced echelon form, the determinant will only have one outcome.
karush
Gold Member
MHB
$\tiny{311.3.2.16}$
Find the determinants where:
$\left|\begin{array}{rrr}a&b&c\\ d&e&f\\5g&5h&5i\end{array}\right| =a\left|\begin{array}{rrr}e&f \\5h&5i\end{array}\right| -b\left|\begin{array}{rrr}d&f \\5g&5i\end{array}\right| +c\left|\begin{array}{rrr}d&e\\5g&5h\end{array}\right|=$

ok before I proceed on
just want see if this is correct
not sure why they thru the 5's in there

This is correct, though I don't like the use of the "i." (Complex numbers and all.)

If you want to get rid of the 5's:
$$\displaystyle \left | \begin{matrix} a & b & c \\ d & e & f \\ 5g & 5h & 5i \end{matrix} \right | = 5 \left | \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right |$$

-Dan

yeah, however I didn't know complex numbers were used in an matrix

$a\left|\begin{array}{rrr}e&f \\5h&5i\end{array}\right| -b\left|\begin{array}{rrr}d&f \\5g&5i\end{array}\right| +c\left|\begin{array}{rrr}d&e\\5g&5h\end{array}\right|$
$=a(e5i-5hf)-b(d5i-5gf)+c(d5h-5ge)$
distirbute
$ae5i-a5hf-bd5i+b5gf+cd5d-c5ge$
rewrite
$5(aei-a5f-bdi+bgf+cdd-cge)$
hopefully,,, I quess the purpose of this was to show that 5 is a scaler
no book answer so not sure how to cross check this

thot I would throw in this question true or false

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row

ok but isn't there just one form of RREF possoble if can be derived? there are multiple ways to reduce it but only one outcome

First, any numbers, including complex numbers, can appear in a matrix.

Second, row reduction of a matrix does NOT preserve its determinant. For example, factoring a number out of an entire row (or column) divides the determinant by that number. That is why, when Topsquark factored the "5"out of the bottom row, he multiplied the determinant by 5.

Swapping two rows, multiplies the determinant by -1.

Finally, adding a multiple of one row to another does not change the determinant.

## 1. What is a determinant?

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

## 2. How do you find the determinant of a matrix with variables?

To find the determinant of a matrix with variables, you can use the method of expansion by minors or the method of cofactors. These methods involve breaking down the matrix into smaller matrices and using algebraic operations to find the determinant.

## 3. What is the purpose of finding the determinant?

The determinant is used in various mathematical applications, such as solving systems of linear equations, calculating the inverse of a matrix, and determining the area or volume of a shape in multi-dimensional space.

## 4. Can the determinant be negative?

Yes, the determinant can be negative. The sign of the determinant depends on the number of row swaps that are performed during the calculation. If there is an odd number of row swaps, the determinant will be negative.

## 5. Is there a shortcut to finding the determinant?

There is no universal shortcut to finding the determinant, as it depends on the size and complexity of the matrix. However, some matrices have special properties that can be used to simplify the calculation, such as diagonal matrices or matrices with many zeros.

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