Find the determinant of the matrix using Gaussian elimination: \begin {bmatrix} 0&1&2 \\-1&1&3...

Question:

Find the determinant of the matrix using Gaussian elimination: {eq}\begin {bmatrix} 0&1&2 \\-1&1&3 \\2&-2&0 \end {bmatrix} {/eq}

Gaussian Elimination

Gaussian elimination is a technique to solve a system of linear equations by eliminating the unknowns from the equations in such a manner that the matrix of the system is an upper triangular matrix.

We can use this technique to find the determinant of a matrix, by finding an equivalent matrix which is upper triangular.

We need to know how the elementary row operations change the determinant, for example, swapping two rows multiplies the determinant by negative one, and multiplying a row by a nonzero scalar multiplies the determinant by the same scalar.

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

Answer and Explanation:

To find the determinant of the square matrix {eq}\displaystyle \begin{align} A=\begin {bmatrix} 0&1&2 \\-1&1&3 \\2&-2&0 \end {bmatrix} \end{align} {/eq} using Gaussian elimination,

we will do row operations on the matrix and keep track of swapping rows or multiplying a row with a scalar.

We will use the notation {eq}\displaystyle R_i {/eq} for the {eq}\displaystyle i- {/eq}th row and describe the operations on the rows on top of the equivalent symbol.


{eq}\displaystyle \begin{align} det(A)=\left|\begin{array}{ccc} 0 & 1 &2\\ -1&1&3 \\ 2&-2&0 \end{array}\right| \overset{R_1 \text{ swap with }R_3}{\iff} -\left|\begin{array}{ccc} 2&-2&0\\ -1&1&3 \\ 0 & 1 &2 \end{array}\right| \overset{R_2+\frac{1}{2}\cdot R_1}{\iff} -\left|\begin{array}{ccc} 2&-2&0\\ 0&0&3 \\ 0 & 1 &2 \end{array}\right|\overset{R_2 \text{ swap with }R_3}{\iff} \left|\begin{array}{ccc} 2&-2&0\\ 0 & 1 &2\\ 0&0&3 \end{array}\right| =2\cdot 1\cdot 3=\boxed{6}. \end{align} {/eq}


We obtained an upper triangular matrix, meaning we have zeros below the diagonal terms, so the determinant is the product of the diagonal terms.

Also, when we interchanged two rows, the determinant is multiplied with negative 1 and since we interchanged twice two rows, the determinant did not change.


Learn more about this topic:

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Inconsistent and Dependent Systems: Using Gaussian Elimination

from Algebra II Textbook

Chapter 10 / Lesson 8
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