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Chapter 3: Problem 24

Use expansion by cofactors to find the determinant of the matrix. $$\left[\begin{array}{rrr} -0.4 & 0.4 & 0.3 \\ 0.2 & 0.2 & 0.2 \\ 0.3 & 0.2 & 0.2 \end{array}\right]$$

Short Answer

Expert verified
The determinant of the given 3x3 matrix is 0.

Step by step solution

01

Choose a Row or Column

Choose a row or column from the matrix to carry out cofactor expansion. Here, the first row is chosen: (-0.4, 0.4, 0.3).
02

Calculate Minors for Each element

Calculate minors for each element from the chosen row. A minor is a determinant of the square set of numbers that is left when the corresponding row and column of a certain element are removed. \n\nMinor of -0.4 is: \(\left[ \begin{array}{cc} 0.2 & 0.2 \ 0.2 & 0.2 \end{array} \right]\).\nMinor of 0.4 is: \(\left[ \begin{array}{cc} 0.2 & 0.2 \ 0.3 & 0.3 \end{array} \right]\).\nMinor of 0.3 is: \(\left[ \begin{array}{cc} 0.2 & 0.2 \ 0.3 & 0.2 \end{array} \right]\).
03

Find the Determinant of the Minors

Calculate the determinant of each minors from step 2. The determinant of a 2x2 matrix \(\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\) can be calculated as (a*d - b*c). \n\nDeterminant for Minor of -0.4: 0.2*0.2 - 0.2*0.2 = 0\nDeterminant for Minor of 0.4: 0.2*0.2 - 0.3*0.2 = -0.02\nDeterminant for Minor of 0.3: 0.2*0.2 - 0.3*0.2 = -0.02
04

Calculate the Cofactors

Calculate the cofactors for each element of chosen row. The cofactor is calculated by multiplying the minor by \(((-1)^{summation of the row and column numbers})\)\n\nCofactor of -0.4: (-0.4) * Minor= (-0.4) * 0 = 0\nCofactor of 0.4: (-0.4) * Minor = (-0.4) * (-0.02) = 0.008\nCofactor of 0.3: (0.3) * Minor = (0.3) * (-0.02) = -0.006
05

Find the Determinant of Original Matrix

The determinant of the original matrix is calculated by summing up the products of the elements of chosen row and their corresponding cofactor.\n\nDeterminant : summation(element[i]*Cofactor[i] )= -0.4*0 + 0.4*0.008 + 0.3*(-0.006) = 0

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Most popular questions from this chapter

Use a graphing utility or computer software program with matrix capabilities to find the eigenvalues of the matrix. Then find the corresponding eigenvectors. $$\left[\begin{array}{rrrr} 1 & 0 & 2 & 3 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & -1 & 3 & 1 \end{array}\right]$$

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 the volume of the tetrahedron having the given vertices. $$(1,0,0),(0,1,0),(0,0,1),(1,1,1)$$

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{array}{r} x_{1}+2 x_{2}=5 \\ -x_{1}+x_{2}=1 \end{array}$$

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{array}{l} 3 x_{1}+4 x_{2}=-2 \\ 5 x_{1}+3 x_{2}=4 \end{array}$$
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