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Chapter 3: Problem 43

Find the determinant of the triangular matrix. $$\left[\begin{array}{rrrr} 5 & 8 & -4 & 2 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & -1 \end{array}\right]$$

Short Answer

Expert verified
The determinant of the given triangular matrix is 0.

Step by step solution

01

Identify the Type of Matrix

The given matrix is an upper triangular matrix as all the elements below its main diagonal are zero.
02

Identify the Diagonal Elements

The diagonal elements are the elements from the top left to the bottom right of the matrix. For the given matrix, the diagonal elements are 5, 0, 2, -1.
03

Compute the Determinant

The determinant of a triangular matrix is the product of its diagonal elements. So, multiply the diagonal elements together: \( 5 \times 0 \times 2 \times (-1) = 0 \).

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Most popular questions from this chapter

Find the volume of the tetrahedron having the given vertices. $$(0,0,0),(0,2,0),(3,0,0),(1,1,4)$$

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{lll} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \end{array}\right]$$

Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{aligned} 3 x_{1}-2 x_{2}+9 x_{3}+4 x_{4} &=35 \\ -x_{1} \quad-9 x_{3}-6 x_{4}=-17 \\ 2 x_{3}+x_{4} =5 \\ 2 x_{1}+2 x_{2}\quad\quad +8 x_{4}=-4 \end{aligned}$$

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{ll} 2 & 1 \\ 3 & 0 \end{array}\right]$$

Find the volume of the tetrahedron having the given vertices. $$(3,-1,1),(4,-4,4),(1,1,1),(0,0,1)$$
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