Find determinant by row reduction in echelon

In summary, the determinants of the given matrix in echelon form is -18. The steps to get to this form include multiplying $r_1$ by 1 and adding it to $r_2$, multiplying $r_1$ by 2 and adding it to $r_3$, and multiplying $r_2$ by -3 and adding it to $r_3$. Finally, the determinant is found by taking the product of the terms in the diagonal of the echelon form matrix.
  • #1
karush
Gold Member
MHB
3,269
5
$\textsf{a. Find the determinants by row reduction in echelon form.}$
$$\left|
\begin{array}{rrr}
1&5&-6\\ -1&-4&4 \\ -2&-7 & 9
\end{array}
\right|$$
ok i multiplied $r_1$ by 1 and added it to $r_2$ to get
$$\left|
\begin{array}{rrr}
1&5&-6\\ 0&1&-2 \\ -2&-7 & 9
\end{array}
\right|$$
but how do you get $0 \,0 \, r_3 c_3$
so it will be in echelon form?
the book answer is $3$
 
Physics news on Phys.org
  • #2
karush said:
$\textsf{a. Find the determinants by row reduction in echelon form.}$
$$\left|
\begin{array}{rrr}
1&5&-6\\ -1&-4&4 \\ -2&-7 & 9
\end{array}
\right|$$
ok i multiplied $r_1$ by 1 and added it to $r_2$ to get
$$\left|
\begin{array}{rrr}
1&5&-6\\ 0&1&-2 \\ -2&-7 & 9
\end{array}
\right|$$
but how do you get $0 \,0 \, r_3 c_3$
so it will be in echelon form?
the book answer is $3$

multiply $r_1$ by 2 and add to $r_3$ ...

$\begin{vmatrix}
1 & 5 & -6\\
0 & 1 & -2 \\
0 & 3 & -3
\end{vmatrix}$

multiply $r_2$ by -3 and add to $r_3$ ...

$\begin{vmatrix}
1 & 5 & -6\\
0 & 1 & -2 \\
0 & 0 & 3
\end{vmatrix}$
 
  • #3
one more

$\textsf{a. Find the determinants by row reduction in echelon form.}$
$$\left|
\begin{array}{rrr}
1&5&-3\\ 3&-3&3 \\ 2&13 &-7
\end{array}
\right|$$
$\textsf{ multiplied $r_1$ by $-2$ and added to $r_3$} $
$$\left|
\begin{array}{rrr}
1&5&-3\\ 3&-3&3 \\ 0&3 &-1
\end{array}
\right|$$
$\textsf{ multiplied $r_1$ by $6$ and added to $r_2$} $
$$\left|
\begin{array}{rrr}
1&5&-3\\ 0&0&-6 \\ 0&3 &-1
\end{array}
\right|$$
$\textsf{ exchange $r_3$ and $r_2$ change sign} $
$$-\left|
\begin{array}{rrr}
1&5&-3\\ 0&3&-1 \\ 0&0 &-6
\end{array}
\right|=18$$
 
Last edited:
  • #4
karush said:
one more

$\textsf{a. Find the determinants by row reduction in echelon form.}$
$$\left|
\begin{array}{rrr}
1&5&-3\\ 3&-3&3 \\ 2&13 &-7
\end{array}
\right|$$
$\textsf{ multiplied $r_1$ by $-2$ and added to $r_3$} $
$$\left|
\begin{array}{rrr}
1&5&-3\\ 3&-3&3 \\ 0&3 &-1
\end{array}
\right|$$
$\textsf{ multiplied $r_1$ by $\color{red}{6}$ and added to $r_2$} $ ?
$$\left|
\begin{array}{rrr}
1&5&-3\\ 0&0&-6 \\ 0&3 &-1
\end{array}
\right|$$
$\textsf{ exchange $r_3$ and $r_2$ change sign} $
$$-\left|
\begin{array}{rrr}
1&5&-3\\ 0&3&-1 \\ 0&0 &-6
\end{array}
\right|=18$$

...

$\begin{bmatrix}
1 & 5 &-3 \\
3 & -3 &3 \\
2 & 13 & -7
\end{bmatrix}$

$-2r_1 + r_3 \to r_3$

$\begin{bmatrix}
1 & 5 &-3 \\
3 & -3 &3 \\
0 & 3 & -1
\end{bmatrix}$

$-3r_1 + r_2 \to r_2$

$\begin{bmatrix}
1 & 5 &-3 \\
0 & -18 &12 \\
0 & 3 & -1
\end{bmatrix}$

$\dfrac{1}{6}r_2 + r_3 \to r_3$

$\begin{bmatrix}
1 & 5 &-3 \\
0 & -18 &12 \\
0 & 0 & 1
\end{bmatrix}$

determinant of an upper triangular matrix is the product of the terms in the diagonal ...

$(1)(-18)(1) = -18$
 

FAQ: Find determinant by row reduction in echelon

1. What is the purpose of finding the determinant by row reduction in echelon?

Finding the determinant by row reduction in echelon form is a method used to solve for the determinant of a square matrix. It allows us to simplify the matrix and make it easier to calculate the determinant, which is a valuable tool in various areas of mathematics and science.

2. How do you perform row reduction in echelon form?

To perform row reduction in echelon form, we use a series of elementary row operations, such as swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. The goal is to transform the matrix into a triangular form, where all elements below the main diagonal are zero.

3. Can you find the determinant by row reduction for any size of matrix?

Yes, the method of finding the determinant by row reduction in echelon form can be applied to any square matrix, regardless of its size. However, the larger the matrix, the more complex and time-consuming the calculations may become.

4. What is the significance of the determinant in mathematics?

The determinant is a useful tool in various areas of mathematics, including linear algebra, calculus, and differential equations. It can be used to determine the invertibility of a matrix, the volume of a parallelepiped formed by the matrix's column vectors, and the solutions to systems of linear equations.

5. Are there any limitations or drawbacks to using row reduction in echelon form to find the determinant?

While row reduction in echelon form is an effective method for finding the determinant, it can be time-consuming and may involve complex calculations, especially for larger matrices. Additionally, it may not be the most efficient method for finding the determinant of matrices with many non-zero entries.

Back
Top