Use Gaussian Elimination to find the determinant of the...

Use Gaussian Elimination to find the determinant of the following matrices: (a) $\left(\begin{array}{rr}2 & -1 \\ -4 & 3\end{array}\right)$, (b) $\left(\begin{array}{rrr}0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0\end{array}\right)$, (c) $\left(\begin{array}{rrr}1 & 2 & 3 \\ 2 & 5 & 8 \\ 3 & 8 & 10\end{array}\right)$, (d) $\left(\begin{array}{rrr}0 & 1 & -1 \\ -2 & 1 & 3 \\ 2 & 7 & -8\end{array}\right)$ (e) $\left(\begin{array}{rrrr}5 & -1 & 0 & 2 \\ 0 & 3 & -1 & 5 \\ 0 & 0 & -4 & 2 \\ 0 & 0 & 0 & 3\end{array}\right)$, (f) $\left(\begin{array}{rrrr}1 & -2 & 1 & 4 \\ 2 & -4 & 0 & 0 \\ 3 & -4 & 2 & 5 \\ 0 & 2 & -4 & -9\end{array}\right)$, (g) $\left(\begin{array}{rrrrr}1 & -2 & 1 & 4 & -5 \\ 1 & 1 & -2 & 3 & -3 \\ 2 & -1 & -1 & 2 & 2 \\ 5 & -1 & 0 & 5 & 5 \\ 2 & 2 & 0 & 4 & -1\end{array}\right)$

Use Gaussian Elimination to find the determinant of the following matrices:
(a) $\left(\begin{array}{rr}2 & -1 \\ -4 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rrr}0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 2 & 3 \\ 2 & 5 & 8 \\ 3 & 8 & 10\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & 1 & -1 \\ -2 & 1 & 3 \\ 2 & 7 & -8\end{array}\right)$
(e) $\left(\begin{array}{rrrr}5 & -1 & 0 & 2 \\ 0 & 3 & -1 & 5 \\ 0 & 0 & -4 & 2 \\ 0 & 0 & 0 & 3\end{array}\right)$,
(f)
$\left(\begin{array}{rrrr}1 & -2 & 1 & 4 \\ 2 & -4 & 0 & 0 \\ 3 & -4 & 2 & 5 \\ 0 & 2 & -4 & -9\end{array}\right)$,
(g)
$\left(\begin{array}{rrrrr}1 & -2 & 1 & 4 & -5 \\ 1 & 1 & -2 & 3 & -3 \\ 2 & -1 & -1 & 2 & 2 \\ 5 & -1 & 0 & 5 & 5 \\ 2 & 2 & 0 & 4 & -1\end{array}\right)$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Upper Triangular Matrices

An upper triangular matrix is one in which all the entries below the main diagonal are zero. The determinant of an upper triangular matrix is simply the product of its diagonal entries, a property that greatly simplifies determinant calculations after Gaussian elimination has been used to transform the matrix.

Gaussian Elimination

This is a method used to simplify matrices by applying a sequence of row operations to convert a matrix into an upper triangular form. It facilitates the computation of determinants, solving systems of linear equations, and finding inverses by systematically eliminating variables from equations.

Elementary Row Operations and Their Impact on the Determinant

Elementary row operations include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. Each of these operations affects the determinant in specific ways: swapping rows changes the sign of the determinant, multiplying a row by a scalar multiplies the determinant by that scalar, while adding a multiple of one row to another leaves the determinant unchanged.

Transcript

Please give Ace some feedback